On nested code pairs from the Hermitian curve
OOn nested code pairs fromthe Hermitian curve
René Bødker Christensen and Olav GeilDepartment of Mathematical Sciences, Aalborg University, Denmark. {rene,olav}@math.aau.dk
Abstract
Nested code pairs play a crucial role in the construction of ramp secret sharingschemes [15] and in the CSS construction of quantum codes [14]. The importantparameters are (1) the codimension, (2) the relative minimum distance of thecodes, and (3) the relative minimum distance of the dual set of codes. Givenvalues for two of them, one aims at finding a set of nested codes having para-meters with these values and with the remaining parameter being as large aspossible. In this work we study nested codes from the Hermitian curve. For nottoo small codimension, we present improved constructions and provide closedformula estimates on their performance. For small codimension we show howto choose pairs of one-point algebraic geometric codes in such a way that oneof the relative minimum distances is larger than the corresponding non-relativeminimum distance.
Keywords:
Algebraic geometric code, Asymmetric quantum code, Hermitiancurve, ramp secret sharing, relative minimum distance
In this paper we study improved constructions of nested code pairs fromthe Hermitian curve. Here the phrase ‘improved construction’ refers tooptimizing those parameters important for the corresponding linear rampsecret sharing schemes as well as stabilizer asymmetric quantum codes. Ourwork is a natural continuation of [7], where improved constructions of nestedcode pairs were defined from Cartesian product point sets. The analysis inthe present paper includes a closed formula estimate on the dimension oforder bound improved Hermitian codes, which is of interest in its own right,i.e. also without the above mentioned applications.A linear ramp secret sharing scheme is a cryptographic method to en-code a secret message in F (cid:96)q into n shares from F q . These shares are thendistributed among a group of n parties and only specified subgroups are ableto reconstruct the secret. A secret sharing scheme is characterized by its1 a r X i v : . [ c s . I T ] J u l rivacy number t and its reconstruction number r . The first is defined as thelargest number such that no subgroup of this size can obtain any informationon the secret. The second is defined to be the smallest number such thatany subgroup of this size can reconstruct the entire secret. A linear rampsecret sharing scheme can be understood as the following coset construction.Consider linear codes C ⊂ C ⊆ F nq . Let { (cid:126)b , . . . ,(cid:126)b k } be a basis for C and extend it to a basis { (cid:126)b , . . . ,(cid:126)b k ,(cid:126)b k +1 , . . . ,(cid:126)b k + (cid:96) } for C . Here, of course, (cid:96) is the codimension of C and C . Choose elements a , . . . , a k uniformlyand independent at random and encode the secret (cid:126)s = ( s , . . . , s (cid:96) ) as thecodeword (cid:126)c = ( c , . . . , c n ) = a (cid:126)b + · · · + a k (cid:126)b k + s (cid:126)b k +1 + · · · + s (cid:96) (cid:126)b k + (cid:96) . Thenuse c , . . . , c n as the shares. The crucial parameters for the constructionare the codimension of the pair of nested codes and their relative minimumdistances d ( C , C ) and d ( C ⊥ , C ⊥ ) . Recall that these are defined as d ( C , C ) = min { w H ( (cid:126)c ) | (cid:126)c ∈ C \ C } and similar for the dual codes. The following well-known theorem (seefor instance [15]) shows how to determine the privacy number and thereconstruction number. Theorem 1.
Given F q -linear codes C ⊂ C of length n and codimension (cid:96) , the corresponding ramp secret sharing scheme encodes secrets (cid:126)s ∈ F (cid:96)q intoa set of n shares from F q . The privacy number equals t = d ( C ⊥ , C ⊥ ) − ,and the reconstruction number is r = n − d ( C , C ) + 1 . A linear q -ary asymmetric quantum error-correcting code is a q k dimen-sional subspace of the Hilbert space C q n where error bases are defined byunitary operators Z and X , the first representing phase-shift errors, and thesecond representing bit-flip errors [2, 21, 14]. In [13] it was identified that insome realistic models phase-shift errors occur more frequently than bit-fliperrors, and the asymmetric codes were therefore introduced [13, 19, 5, 16, 25]to balance the error correcting ability accordingly. For such codes we writethe set of parameters as [[ n, k, d z /d x ]] q where d z is the minimum distancerelated to phase-shift errors and d x is the minimum distance related tobit-flip errors. The CSS construction transforms a pair of nested classicallinear codes C ⊂ C ⊆ F nq into an asymmetric quantum code. From [19]we have Theorem 2.
Consider linear codes C ⊂ C ⊆ F nq . Then the correspondingasymmetric quantum code defined using the CSS construction has parameters [[ n, (cid:96) = dim C − dim C , d z /d x ]] q where d z = d ( C , C ) and d x = d ( C ⊥ , C ⊥ ) . d ( C , C ) > d ( C ) or d ( C ⊥ , C ⊥ ) > d ( C ⊥ ) are calledimpure, and they are desirable due to the fact that one can take advantage ofthis property in connection with the error-correction. More precisely, one cantolerate (cid:98) ( d ( C , C ) − / (cid:99) phase-shift errors and (cid:98) ( d ( C ⊥ , C ⊥ ) − / (cid:99) bit-flip errors, respectively, but in the decoding algorithms it is only necessaryto correct up to (cid:98) ( d ( C ) − / (cid:99) and (cid:98) ( d ( C ⊥ ) − / (cid:99) errors, respectively.Despite this observation, only few impure codes have been presented in theliterature.With the above two applications in mind, the challenge is to find nestedcodes C ⊂ C such that two of the parameters (cid:96) , d ( C , C ) , d ( C ⊥ , C ⊥ ) attain given prescribed values, and the remaining parameter is as largeas possible. In this paper we analyse two good constructions from theHermitian function field. In the first we consider code pairs such that C isan order bound improved primary code [1, 10] and such that C is the dualof an order bound improved dual code [12]. Considering in this case theminimum distances rather than the relative distances is no restriction dueto the optimized choice of codes – the minimum distances d ( C ) and d ( C ⊥ ) being so good that there is essentially no room for d ( C , C ) > d ( C ) or d ( C ⊥ , C ⊥ ) > d ( C ⊥ ) to hold. For this construction to work, the codimensioncannot be very small. For small codimension when d ( C ) and d ( C ⊥ ) are farfrom each other we then show how to choose ordinary one-point algebraicgeometric codes such that one of the relative distances becomes largerthan the corresponding ordinary minimum distance. In particular, thisconstruction leads to impure asymmetric quantum codes.The paper is organized as follows. In Section 2 we collect materialfrom the literature on how to determine parameters of primary and dualcodes coming from the Hermitian curve, and we introduce the order boundimproved codes . In Section 3 we establish closed formula lower bounds onthe dimension of order bound improved Hermitian codes of any designedminimum distance. We then continue in Section 4 by determining thepairs ( δ , δ ) ∈ { , . . . , n } × { , . . . , n } for which the order bound improvedprimary code C of designed distance δ contains C , the dual of an orderbound improved dual code of designed distance δ . This and the informationfrom Section 3 is then translated into information on improved nested codepairs of not too small codimension in Section 5. Next, in Section 6 wedetermine parameters of nested one-point algebraic geometric code pairs ofsmall codimension for which one of the relative distances is larger than the This section also contains a collective treatment of the order bounds for generalprimary as well as dual (improved) one-point algebraic geometric codes which may notbe easy to find in the literature.
Given an algebraic function field over a finite field, let = P , . . . , P n , Q berational places. By H ( Q ) we denote the Weierstrass semigroup of Q , andwe write H ∗ ( Q ) = { λ ∈ H ( Q ) | C L ( D, λQ ) (cid:54) = C L ( D, ( λ − Q ) } where D = P + · · · + P n . Recall that the dual code of C L ( D, λQ ) is written C Ω ( D, λQ ) . The order bound [12, 4] then tells us that if (cid:126)c ∈ C Ω ( D, ( λ − Q ) \ C Ω ( D, λQ ) (which can only happen if λ ∈ H ∗ ( Q ) ), then the Hamming weight of (cid:126)c satisfies w H ( (cid:126)c ) ≥ µ ( λ ) (1)where µ ( λ ) = { η ∈ H ( Q ) | λ − η ∈ H ( Q ) } . The similar bound for the primary case [8, 1, 10] tells us that if (cid:126)c ∈ C L ( D, λQ ) \ C L ( D, ( λ − Q ) , S then w H ( (cid:126)c ) ≥ σ ( λ ) (2)where σ ( λ ) = { η ∈ H ∗ ( Q ) | η − λ ∈ H ( Q ) } . Besides implying that d ( C L ( D, λQ )) ≥ min { σ ( γ ) | ≤ γ ≤ λ, γ ∈ H ∗ ( Q ) } d ( C Ω ( D, λQ )) ≥ min { µ ( γ ) | λ < γ, γ ∈ H ∗ ( Q ) } , (3)which are both as strong as the Goppa bound, it tells us that for (cid:15), λ ∈ H ∗ ( Q ) with (cid:15) < λ it holds that d ( C L ( D, λQ ) , C L ( D, (cid:15)Q )) ≥ min { σ ( γ ) | (cid:15) < γ ≤ λ, γ ∈ H ∗ ( Q ) } , (4)4nd similarly d ( C Ω ( D, (cid:15)Q ) , C Ω ( D, λQ )) ≥ min { µ ( γ ) | (cid:15) < γ ≤ λ, γ ∈ H ∗ ( Q ) } . (5)Furthermore, for i ∈ H ∗ ( Q ) let f i ∈ L ( iQ ) \L (( i − Q ) . Then we obtainthe improved primary code (cid:101) E ( δ ) = Span { ( f i ( P ) , . . . , f i ( P n )) | σ ( i ) ≥ δ } , which clearly has minimum distance at least δ and highest possible dimensionfor a primary code with that designed distance. Similarly, the improveddual code (cid:101) C ( δ ) = (cid:0) Span { ( f i ( P ) , . . . , f i ( P n )) | µ ( i ) < δ } (cid:1) ⊥ has minimum distance at least δ and again the highest possible dimensionfor a dual code with that designed distance.Turning to the Hermitian curve x q +1 − y q − y over F q where q is aprime power, it is well-known that the corresponding function field hasexactly q + 1 rational places P , . . . , P q , Q . Choosing n = q one obtains H ( Q ) = (cid:104) q, q + 1 (cid:105) and H ∗ ( Q ) = { iq + j ( q + 1) | ≤ i ≤ q − , ≤ j ≤ q − } . (6)In [23] it was shown that C L ( D, λQ ) = C Ω ( D, ( q + q − q − − λ ) Q ) (7)for any λ ∈ H ∗ ( Q ) , and the minimum distance was established for di-mensions up to a certain value. The minimum distance for the remainingdimensions was then settled in [26]. In the present paper we shall needimproved code constructions, and we will in some cases also be occupiedwith the relative distances rather than minimum distances. To this end werecall material from [8] on the functions µ and σ – stated there in the moregeneral case of norm-trace curves, but adapted here to the Hermitian case. Proposition 3.
Consider the Hermitian curve. For iq + j ( q + 1) ∈ H ∗ ( Q ) we have σ ( iq + j ( q + 1)) = (cid:26) q − iq − j ( q + 1) if ≤ i < q − q ( q − i )( q − j ) if q − q ≤ i ≤ q − , (8) and µ (cid:0) ( q − − i ) q + ( q − − j )( q + 1) (cid:1) = σ (cid:0) iq + j ( q + 1) (cid:1) . For each λ ∈ H ∗ ( Q ) there exists a word (cid:126)c ∈ (cid:0) C L ( D, λQ ) \ C L ( D, ( λ − Q ) (cid:1) ∩ (cid:101) E ( σ ( λ )) having Hamming weight equal to σ ( λ ) . roof. Given a numerical semigroup Λ with finitely many gaps and anelement λ ∈ Λ , we know from [12, Lem. 5.15] that (cid:0) Λ \ ( λ + Λ) (cid:1) = λ . As H ∗ ( Q ) = q we therefore obtain σ ( iq + j ( q + 1)) ≥ q − ( iq + j ( q + 1)) .On the other hand, it is clear that σ ( iq + j ( q + 1)) ≥ ( q − i )( q − j ) by thedefinition of σ . Taking the maximum between these two expressions, weobtain the right hand side of (8). That these estimates on σ are the truevalues and that the last part of the proposition holds true both follow as aconsequence of [8, Lem. 4]. The details of applying [8, Lem. 4] are left forthe reader. Finally, the relation between µ and σ is a consequence of H ∗ ( Q ) being a box in the parameters i and j , see (6).In Appendix A we list a series of lemmas which all follow as corollaries toProposition 3 and which will be needed in Sections 3 and 4.Throughout the rest of the paper we restrict to considering codes derivedfrom the Hermitian curve, and we always assume the length to be n = q .From Proposition 3 we see that the bound (4) on the relative distance of C L ( D, (cid:15)Q ) ⊂ C L ( D, λQ ) is sharp. A similar remark then holds for thebound (5) on the dual codes due to (7). Finally, we observe from [8, Sec. 4]that (cid:101) E ( δ ) = (cid:101) C ( δ ) (9)holds. Proposition 3 therefore not only gives us the true value of theminimum distance of the improved primary codes (without loss of generalitywe may assume δ = σ ( λ ) for some λ ∈ H ∗ ( Q ) ), but also does it for theimproved dual codes.We conclude the section with some information on the cases where theimproved primary codes coincide with one-point algebraic geometric codes. Corollary 4.
For δ > q − q we have (cid:101) E ( δ ) = C L (cid:0) D, ( q − δ ) Q (cid:1) , but C L (cid:0) D, ( q − ( q − q )) Q (cid:1) is strictly contained in (cid:101) E ( q − q ) . This corollary implies that the dimension of (cid:101) E ( δ ) can be determined fromthe usual one-point Hermitian codes whenever δ > q − q . For later referencewe state these dimensions in terms of δ . Proposition 5.
Denote by g = q ( q − / the genus of Hermitian functionfield. If q − q < δ < q − g + 2 , then the dimension of (cid:101) E ( δ ) is given by q − g + 1 − δ . If q − g + 2 ≤ δ ≤ q , we have dim (cid:101) E ( δ ) = a + b (cid:88) s =0 ( s + 1) − max { a, } where q − δ = aq + b ( q + 1) for − q < a < q and ≤ b < q . roof. First, note that in both cases Corollary 4 implies the equality (cid:101) E ( δ ) = C L ( D, ( q − δ ) Q ) . For the first case recall from [24, Cor. II.2.3] that thecode C L ( D, λG ) has dimension λ + 1 − g whenever g − < λ < n . By theassumptions on δ , we have q − δ > q − ( q − g + 2) = 2 g − , meaningthat C L ( D, ( q − δ ) Q ) has dimension q − δ + 1 − g . By the observation inthe beginning of the proof, the same holds true for (cid:101) E ( δ ) .To prove the second case, observe that the dimension of C L ( d, ( q − δ ) Q ) is exactly the number of elements λ in H ∗ ( Q ) with λ ≤ q − δ . By (6) suchelements have the form λ = iq + j ( q + 1) , but equivalently we can write λ = i (cid:48) q + j where i (cid:48) = i + j . By the division algorithm this representation isunique for ≤ j < q . For i (cid:48) q + j to satisfy the requirements of (6), we alsorequire i (cid:48) − j = i ≥ . That is, H ∗ ( Q ) contains the integers whose quotientsmodulo q are at least their remainders modulo q .Writing q − δ = ( a + b ) q + b with ≤ b < q using the division algorithm,the number of elements in H ∗ ( Q ) less than ( a + b +1) q is given by (cid:80) a + bs =0 ( s +1) .If b ≥ a + b , which happens only if a ≤ , this number is also the number ofelements with value at most q − δ . Otherwise, the count includes b − ( a + b ) = a elements of H ∗ ( Q ) that are greater than q − δ . Hence, subtracting max { a, } gives the desired count in both cases.It remains to establish information on the dimension of (cid:101) E ( δ ) for δ ≤ q − q since the improved codes differ from the usual Hermitian codes in this case.This subject is treated in the next section. As explained in the previous section, the dimension of (cid:101) E ( δ ) can be determ-ined from well-known methods as long as δ > q − q . In this section wepresent closed formula lower bounds on the dimension in the remainingcases. We start with an important lemma. Lemma 6.
Let δ ≤ q . The number of integer points ( x, y ) ∈ { q − q, . . . , q − } × { , . . . , q − } with ( q − x )( q − y ) ≥ δ is at least q − (cid:4) δ + δ ln( q /δ ) (cid:5) . (10) If δ < q , then the number of integer points is at least q − (cid:98) δ + δ ln( δ ) (cid:99) , (11) which is stronger than (10). roof. The number of integer points in the given Cartesian product is atleast that of the volume of { ( x, y ) ∈ [ q − q, q ] × [0 , q ] | ( q − x )( q − y ) ≥ δ } , which equals (cid:90) q − δq q − q (cid:90) q − δq − x dy dx = q ( q − δq − q + q ) + (cid:90) q − qq − δq δq − x dx = q − δ − δ [ln( z )] q δq = q − δ − δ ln( q /δ ) , where we used the substitution z = q − x . Since the number of integerpoints is integral, we obtain the bound (cid:100) q − δ − δ ln( q /δ ) (cid:101) , which is thesame as (10).If δ < q , then the number of integer points is at least the combinedvolumes of { ( x, y ) ∈ [ q − q, q ] × [0 , q ] | x ≤ q − δ or y ≤ q − δ } and { ( x, y ) ∈ [ q − δ, q ] × [ q − δ, q ] | ( q − x )( q − y ) ≥ δ } . The first mentioned volume equals q − δ . The latter volume is (cid:90) q − q − δ (cid:90) q − δq − x q − δ dy dx = (cid:90) q − q − δ (cid:18) δ − δq − x (cid:19) dx = δ ( δ − − δ [ln( z )] δ = δ ( δ − − δ ln( δ ) . Adding up the two volumes, we obtain (11) by applying the ceiling functionas above.The dimension of the improved codes of designed distance at most q − q iscovered by the following two propositions. Recall from (9) that the equality (cid:101) C ( δ ) = (cid:101) E ( δ ) holds for codes defined from the Hermitian function field.Hence, the stated formulas for primary codes also hold for the dual codes.8 roposition 7. Given q < δ ≤ q − q write q − δ = q − q + aq + b ( q + 1) where − q < a < q and ≤ b < q .If < a , then dim( (cid:101) E ( δ )) ≥ q − δ − g + 1 − a + b (cid:88) s =0 ( s + 1) + a + q − (cid:4) δ + δ ln( q /δ ) (cid:5) . If a ≤ , then dim( (cid:101) E ( δ )) ≥ q − δ − g + 1 − a + b (cid:88) s =0 ( s + 1) + q − (cid:4) δ + δ ln( q /δ ) (cid:5) . Proof.
Let g = q ( q − / be the number of gaps in H ( Q ) , i.e. the genusof the function field. As is well-known, for g ≤ λ < q − the number of (cid:15) ∈ H ∗ ( Q ) with (cid:15) ≤ λ equals λ − g + 1 . Therefore, by choosing λ = q − δ the restriction on δ as given in the proposition implies that there are exactly q − δ − g + 1 elements (cid:15) ∈ H ∗ ( Q ) with (cid:15) ≤ q − δ . These elements thensatisfy σ ( (cid:15) ) ≥ δ by Lemma 25. From (8) we see that the additional elementsin H ∗ ( Q ) with σ ( (cid:15) ) ≥ δ must belong to { iq + j ( q + 1) | q − q ≤ i ≤ q − , ≤ j ≤ q − } . (12)Lemma 6 gives an estimate on the total number of elements (cid:15) in (12) with σ ( (cid:15) ) ≥ δ . Adding this number to q − δ − g +1 , we have counted the elements (cid:15) in (12) with (cid:15) ≤ q − δ twice. By using similar arguments as in the proofof Proposition 5, the number of such elements equals (cid:80) a + bs =0 ( s + 1) − a when ≤ a < q , and it equals (cid:80) a + bs =0 ( s + 1) when − q < a < . This proves theproposition. Proposition 8.
Given ≤ δ ≤ q the dimension of the code (cid:101) E ( δ ) satisfies dim( (cid:101) E ( δ )) ≥ q − (cid:98) δ + δ ln( δ ) (cid:99) . Proof.
By Lemma 30 the elements λ ∈ H ∗ ( Q ) which do not satisfy σ ( λ ) ≥ δ must belong to { iq + j ( q + 1) | q − δ ≤ i ≤ q , q − δ ≤ j < q } . The numberof elements in this set having σ ( λ ) < δ is bounded above by (cid:98) δ + δ ln( δ ) (cid:99) byLemma 6. Since the total number of monomials in H ∗ ( Q ) is q , the resultfollows. 9 Inclusion of improved codes
As already mentioned our first construction of improved nested code pairsconsists of choosing (cid:101) C ( δ ) and (cid:101) E ( δ ) such that (cid:101) C ( δ ) ⊥ ⊂ (cid:101) E ( δ ) . To treatthis construction we therefore need a clear picture of the pairs ( δ , δ ) ofminimum distances that imply this inclusion. We establish this in the presentsection. As it turns out, the formulas for σ and µ given in Proposition 3mean that several cases must be considered, and each case is presented asa separate proposition.In the following, quantifiers on λ, ε are considered on the domain H ∗ ( Q ) .Given δ ∈ σ ( H ∗ ( Q )) , define δ max2 to be the maximal value of δ such that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) holds. This inclusion is equivalent to ∀ λ : (cid:2) ( σ ( λ ) < δ ) → ( µ ( λ ) ≥ δ ) (cid:3) . (13)We first observe that if we can find a λ ∈ H ∗ ( Q ) such that (cid:2) ∀ ε > λ : µ ( ε ) ≥ µ ( λ ) (cid:3) ∧ (cid:2) ∀ ε < λ : σ ( ε ) ≥ δ (cid:3) (14)is true, then (13) is also true whenever δ ≤ µ ( λ ) . In particular, wetherefore have δ max2 ≥ µ ( λ ) . (15)On the other hand, we immediately see from (13) that a λ ∈ H ∗ ( Q ) with σ ( λ ) < δ (16)implies the bound δ max2 ≤ µ ( λ ) . (17)In the proofs of each of the following propositions, our strategy thereforeis to determine λ and λ satisfying (14) and (16), respectively, while alsosatisfying µ ( λ ) = µ ( λ ) . From (15) and (17) it then follows that δ max2 = µ ( λ ) . Note, however, that λ and λ need not be distinct. If λ = λ , weshall simply use λ .With this strategy in mind, the following lemmas will prove very useful. Lemma 9.
Let λ = iq + j ( q + 1) ∈ H ∗ ( Q ) , meaning that ≤ i < q and ≤ j < q . In addition, assume that i ≤ q − q , i = q − , or j = 0 . If ε ∈ H ∗ ( Q ) satisfies ε < λ , then σ ( ε ) ≥ σ ( λ ) . roof. For both the cases i < q − q and j = 0 , the claim follows byLemma 26. If i = q − q , Lemma 28 implies that σ ( λ ) = σ (( i + j ) q ) , and any ε ∈ H ∗ ( Q ) satisfying λ > ε > ( i + j ) q has σ ( ε ) ≥ σ (( i + j ) q ) by Lemma 29.This also holds true for ε < ( i + j ) q by the first part of the proof.Finally, for i = q − consider ε = i (cid:48) q + j (cid:48) ( q + 1) ∈ H ∗ ( Q ) with ε < λ . If q − q ≤ i (cid:48) ≤ q − , the claim follows by Lemmas 27 and 29. Otherwise, ε isat most ( q − q − q + ( q − q + 1) = q − q − , meaning that σ ( ε ) ≥ q + 1 by Lemma 26. The proof follows by noting that σ ( λ ) ≤ q . Lemma 10.
Let λ = iq + j ( q + 1) ∈ H ∗ ( Q ) , meaning that ≤ i < q and ≤ j < q . In addition, assume that i ≥ q − or j = 0 . If ε ∈ H ∗ ( Q ) satisfies ε > λ , then µ ( ε ) ≥ µ ( λ ) .Proof. This proof is similar to the one of Lemma 9. Defining the notation λ (cid:48) = ( q − − i ) q + ( q − − j )( q + 1) , the translation of Lemma 25 intoinformation on µ gives µ ( λ ) = n − λ (cid:48) whenever i > q − or j = q − . (18)Additionally, if ε ∈ H ∗ ( Q ) with ε > λ , then ε (cid:48) < λ (cid:48) where the ε (cid:48) is definedin the same way as λ (cid:48) . This implies that µ ( λ ) < µ ( ε ) when λ = iq + j ( q + 1) satisfies (18). This immediately proves the claim for i > q − .For i = q − the µ -equivalent of Lemma 28 gives µ ( λ ) = µ (( i − ( q − − j )) + ( q − q + 1)) , and any elements between have greater µ -value bythe translation of Lemma 29. The remaining elements greater than λ arecovered by the first part of the proof.The last part of the proof is j = 0 and i < q − , which follows the sameprocedure as the last part of the proof of Lemma 9. Proposition 11.
Let ≤ δ ≤ q . Then (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) if and only if δ ≤ q − ( δ − q + 1) .Proof. Let λ (cid:48) = ( q − q + ( q − δ )( q + 1) . We have σ ( λ (cid:48) ) = δ by (8), andLemma 9 implies that σ ( ε ) ≥ δ for all ε < λ (cid:48) . Additionally, Lemmas 27 and29 yield that λ = λ (cid:48) + q + 1 is the smallest element in H ∗ ( Q ) with a strictlysmaller σ -value. Combining this with Lemma 10 applied to λ reveals that λ satisfies (14). However, (16) is satisfied as well since σ ( λ ) < δ . Thus, δ max2 = µ ( λ ) = q − ( δ − q + 1) . Proposition 12.
Let q < δ ≤ q − q . Then (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) if and only if δ ≤ (cid:40) q − q + q − δ + 2 if ≤ b ≤ aq − q − a ( q + 1) if b > a where δ − ( q + 1) = aq + b for a ≥ and ≤ b < q . roof. First, observe that aq + b ≤ ( q − a + ( q − , meaning that a is atmost q − .Assume that b = 0 . Then λ = ( q − − a ) q has σ ( λ ) = δ − , meaningthat it satisfies (16). Note that µ ( λ ) = q − q − aq + 1 , which can berewritten to obtain the claimed expression.If < b ≤ a , we can use λ = ( q − q − − a +( b − q +( q − − ( b − q +1) ,which satisfies (16) since σ ( λ ) = δ − . Here, we see that µ ( λ ) = q − q − aq − b + 1 = q − q + q − δ + 2 . Finally, for b > a we can let λ = ( q − q − q + ( q − − a )( q + 1) with σ ( λ ) = ( a + 1) q + a + 1 < δ . Again, λ satisfies (16). Calculating the valueof µ gives µ ( λ ) = q − q − a ( q + 1) .In all three of the above situations, the element immediately preceding λ in H ∗ ( Q ) is given by λ (cid:48) = λ − , and the reader may verify that σ ( λ (cid:48) ) ≥ δ .In each case applying Lemma 9 to λ (cid:48) implies that σ ( ε ) ≥ δ for all ε < λ .Lemma 10 applied to λ then shows that λ satisfies (14) as well. In conclusion,the specified values of λ satisfy both (14) and (16), and computing eachvalue of µ ( λ ) gives the expression in the proposition. Proposition 13.
Let q − q < δ ≤ q − q + 2 q . Then (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) ifand only if δ ≤ q − q + q + 2 − δ .Proof. Set λ (cid:48) = n − δ and observe that λ (cid:48) ≥ q − q = 4 g where g is thegenus of the Hermitian function field. Thus, λ (cid:48) is a non-gap in H ∗ ( Q ) , and σ ( λ (cid:48) ) = δ by (8). Lemma 9 implies that any smaller element of H ∗ ( Q ) has σ -value at least δ . We see, however, that λ = λ (cid:48) + 1 has σ ( λ ) = δ − ,and it must be the smallest such value. At the same time it meets therequirements of Lemma 10, implying that (14) is fulfilled. As already noted λ satisfies (16) as well, meaning that δ max2 = µ ( λ ) = q − q + q + 2 − δ . Proposition 14.
Let q − q + 2 q < δ ≤ q − q . Then (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) if and only if δ ≤ ( a + 1) q + b + 2 if b < a ( a + 2) q if a ≤ b < q − a + 2) q + 1 if b = q − where q − q − δ = aq + b for a ≥ and ≤ b < q .Proof. First note that aq + b ≤ ( q − q , implying that a is at most q − .Assume that b < a and let λ = ( q + a − ( b + 1)) q + ( b + 1)( q + 1) . This element12atisfies the requirements of Lemma 10, and λ − satisfies the requirementsof Lemma 9. This means that λ fulfils (14). Simultaneously, (16) is metsince σ ( λ ) = δ − . Thus, δ max2 = µ ( λ ) = ( a + 1) q + b + 2 .Let a = b and λ = ( q − q + ( a + 1)( q + 1) . Applying Lemmas 9 and10 to λ − and λ as above, we see that λ satisfies (14). It also meets (16)since σ ( λ ) = δ − . Subsequently, δ max2 = µ ( λ ) = ( a + 2) q .Now, consider a < b < q − and let λ = ( q − q + ( a + 1)( q + 1) and λ = ( a + 1) q + ( q − q + 1) . We can apply both Lemmas 9 and 10 to λ toobtain that it satisfies (14). On the other hand, σ ( λ ) < δ implies that (16)is fulfilled. In addition, µ ( λ ) = µ ( λ ) , which gives δ max2 = µ ( λ ) = ( a + 2) q .The remaining part is b = q − . If this happens, note that λ = ( q + a +1) q has σ ( λ ) = δ − , whereas λ − a + 1) q + ( q − q + 1) has σ ( λ −
1) = δ .By the same arguments as in the first part of the proof, we obtain that δ max2 = µ ( λ ) = ( a + 2) q + 1 . Proposition 15.
Let q − q ≤ δ ≤ q . Then (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) if and onlyif δ ≤ (cid:40) a + 1 if b < aa + 2 if b ≥ a where q − δ = aq + b for a ≥ and ≤ b < q .Proof. Assume first that b < a , and set λ = aq and λ = a ( q + 1) . Thelatter meets the assumptions of Lemmas 9 and 10, meaning that λ satisfies(14). Observe that σ ( λ ) < δ and µ ( λ ) = µ ( λ ) . From this we see that δ max2 = µ ( λ ) = a + 1 .Otherwise, if b ≥ a , let λ = ( a + 1) q , which satisfies (16) by the obser-vation that σ ( λ ) < δ . The element of H ∗ ( Q ) immediately preceding λ is λ (cid:48) = a ( q + 1) , which has σ ( λ (cid:48) ) ≥ δ . As in the previous proofs, applyingLemmas 9 and 10 to λ (cid:48) and λ , respectively, shows that λ fulfils (14). Hence, δ max2 = µ ( λ ) = a + 2 .It is worth noting that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) if and only if every λ ∈ H ∗ ( Q ) with σ ( λ ) < δ also satisfies µ ( λ ) ≥ δ . By Proposition 3 this may be rewrittenas µ ( λ ) < δ implying σ ( λ ) ≥ δ . Hence, the inclusion of codes is symmetricin the sense that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) if and only if (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) .One could expect that this symmetry would show up in Propositions11–15 as well. However, this is not the case since the propositions describethe maximal value of δ such that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) for a given value of δ .Although this implies that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) , there may be a δ (cid:48) > δ suchthat (cid:101) C ( δ (cid:48) ) ⊥ ⊆ (cid:101) E ( δ ) as shown in Example 1 below.13 xample 1. Let q = 4 and set δ = 6 . Then considering the values of σ and µ as given in Table 4 of the Appendix reveals that the greatest possible valueof δ such that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) is given by δ = 48 . By the observationsabove we know that this implies (cid:101) C (6) ⊥ ⊆ (cid:101) E (48) as well. However, inspectingthe tables again will reveal that the (cid:101) C (8) ⊥ is also a subset of (cid:101) E (48) . Noticethat both of these observations agree with the formulas in Propositions 12and 14. Based on our findings in Sections 3 and 4, we are now able to describethe parameters of our first construction of nested code pairs, namely theone where the codimension is not too small. If δ , δ ∈ H ∗ ( Q ) satisfy theconditions in one of the Propositions 11–15, it follows that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) .By the bounds (4) and (5) and the observation following Proposition 3, therelative distance of this code pair is exactly d ( (cid:101) E ( δ )) = δ , and the relativedistance of its dual is d ( (cid:101) C ( δ ) ⊥ ) = δ .For each possible pair of designed distances described in Propositions 11–15, we can combine the dimensions of the usual Hermitian codes withthe dimension bounds of Propositions 7 and 8. This gives bounds on thecodimension, (cid:96) , of (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) . Proposition 16.
Let δ , δ ∈ H ∗ ( Q ) , and δ ≤ q . Further, let δ satisfy theconditions of Proposition 11, meaning that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) . Denote theircodimension by (cid:96) .If δ ≤ q , then (cid:96) ≥ q − (cid:98) δ + δ ln( δ ) (cid:99) − (cid:98) δ + δ ln( δ ) (cid:99) . If q < δ ≤ q − q , then (cid:96) ≥ q + q − g + 1 − (cid:98) δ + δ ln( δ ) (cid:99) − δ − a + b (cid:88) s =0 ( s + 1) − (cid:98) δ + δ ln (cid:0) q /δ (cid:1) (cid:99) + max { a, } where a and b are as in Proposition 7 applied to δ .If q − q < δ < q − g + 2 , then (cid:96) ≥ q − g + 1 − (cid:98) δ + δ ln( δ ) (cid:99) − δ . inally, if q − g + 2 ≤ δ , we have (cid:96) ≥ a + b (cid:88) s =0 ( s + 1) − (cid:98) δ + δ ln( δ ) (cid:99) − max { a, } where q − δ = aq + b ( q + 1) for − q < a < q and ≤ b < q .Proof. By Proposition 8 we have dim (cid:101) E ( δ ) ≥ q − (cid:98) δ + δ ln( δ ) (cid:99) . In eachcase, we can obtain a bound on the codimension (cid:96) by subtracting an upperbound on (cid:101) C ( δ ) ⊥ = q − dim (cid:101) E ( δ ) . In turn, such a bound can be obtainedvia a lower bound on dim (cid:101) E ( δ ) .In the case δ ≤ q the dimension of (cid:101) E ( δ ) can again be bounded byProposition 8. In the second case the bound on dim (cid:101) E ( δ ) follows by Pro-position 7. Proposition 5 delivers the bounds in the two final cases. Proposition 17.
Let δ , δ ∈ H ∗ ( Q ) and q < δ ≤ q − q . Further, let δ satisfy the conditions of Proposition 11, meaning that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) .Denote their codimension by (cid:96) and let a , b be as in Proposition 7 appliedto δ .If δ ≤ q , then (cid:96) ≥ q + q − g + 1 − δ − a + b (cid:88) s =0 ( s + 1) + max { a , }− (cid:98) δ + δ ln( q /δ ) (cid:99) − (cid:98) δ + δ ln( δ ) (cid:99) . If q < δ ≤ q − q , then (cid:96) ≥ q + 2 q − g + 2 − ( δ + δ ) − a + b (cid:88) s =0 ( s + 1) + max { a , }− a + b (cid:88) s =0 ( s + 1) + max { a , } − (cid:98) δ + δ ln( q /δ ) (cid:99) − (cid:98) δ + δ ln( q /δ ) (cid:99) where a and b are as in Proposition 7 applied to δ .Finally, if q − q < δ , then (cid:96) ≥ q + q − g + 2 − ( δ + δ ) − a + b (cid:88) s =0 ( s + 1) − (cid:98) δ + δ ln( q /δ ) (cid:99) + max { a , } . Proof.
We use the same strategy as in the proof of Proposition 16. Abound for the dimension of (cid:101) E ( δ ) can be found in Proposition 7. For15 ≤ q the bound on dim (cid:101) E ( δ ) comes from Proposition 8, and in thecase q < δ ≤ q − q it comes from Proposition 7. In the final case thebound follows from Proposition 5, where we note that δ ≤ q − g + 2 byProposition 12 and the assumption on δ . Proposition 18.
Let δ , δ ∈ H ∗ ( Q ) and q − q < δ < q − g + 2 . Further,let δ satisfy the conditions of Proposition 11, meaning that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) .Denote their codimension by (cid:96) .If δ ≤ q , we have (cid:96) ≥ q − g + 1 − δ − (cid:98) δ + δ ln( δ ) (cid:99) . If q < δ ≤ q − q , then (cid:96) ≥ q + q − g + 2 − ( δ + δ ) − a + b (cid:88) s =0 ( s + 1) − (cid:98) δ + δ ln( q /δ ) (cid:99) + max { a, } where a and b are as in Proposition 7 applied to δ .Finally, for q − q < δ we have (cid:96) = q − δ − δ − g + 2 . Proof.
Again, the the strategy is the same as in the proof of Proposition 16.The exact dimension of (cid:101) E ( δ ) is given by Proposition 5. For δ ≤ q thedimension of (cid:101) E ( δ ) can be bounded by applying Proposition 8, and in thecase q < δ ≤ q − q the bound follows by Proposition 7. For the final casewe note by Proposition 13 that δ < q − q + q + 2 − ( q − q ) = q − g + 2 .Hence, the exact dimension of (cid:101) E ( δ ) is given by the first part of Proposition 5in this case. Proposition 19.
Let δ , δ ∈ H ∗ ( Q ) and q − g + 2 ≤ δ . Further, let δ satisfy the conditions of Proposition 11, meaning that (cid:101) C ( δ ) ⊥ ⊆ (cid:101) E ( δ ) .Denote their codimension by (cid:96) . Then (cid:96) ≥ a + b (cid:88) s =0 ( s + 1) − max { a, } − (cid:98) δ + δ ln( δ ) (cid:99) where q − δ = aq + b ( q + 1) for − q < a < q and ≤ b < q .Proof. The dimension of (cid:101) E ( δ ) is given by the last part of Proposition 5. Toobtain a bound on the maximal value of δ , note that the the minimal valueof q − δ can be written as q − q − q ( q −
2) + ( q − . Proposition 15 nowimplies δ ≤ q . Hence, dim (cid:101) E ( δ ) ≥ q −(cid:98) δ + δ ln( δ ) (cid:99) by Proposition 8.16he application of Theorem 1 or 2 translates Propositions 16–19 into in-formation on improved linear ramp secret sharing schemes and improvedasymmetric quantum codes, respectively. The details are left for the reader. We will now consider a second construction which in general gives nestedcode pairs of smaller codimension than the construction in Section 5. Thisconstruction bears some resemblance to the one given in [7, Sec. IV], butin the setting of Hermitian codes.From the definition of the codes, C L ( D, λ Q ) (cid:40) C L ( D, λ Q ) whenever λ < λ and both λ and λ belongs to H ∗ ( Q ) . Our second construction iscaptured by the following two propositions. Proposition 20.
Let λ = iq + j ( q +1) ∈ H ∗ ( Q ) where i ≤ j < q , and define λ = jq + i ( q + 1) − . Then C L ( D, λ Q ) (cid:40) C L ( D, λ Q ) have codimension (cid:96) = j − i + 1 , and their relative distances satisfy d (cid:0) C L ( D, λ Q ) , C L ( D, λ Q ) (cid:1) = q − λ = d ( C L ( D, λ Q )) , (19) and d (cid:0) C L ( D, λ Q ) ⊥ , C L ( D, λ Q ) ⊥ (cid:1) = ( i + 1)( j + 1) ≥ d (cid:0) C L ( D, λ Q ) ⊥ (cid:1) . (20) The inequality in (20) is strict if and only if i (cid:54) = 0 and j (cid:54) = q − .Proof. The codimension of C L ( D, λ Q ) and C L ( D, λ Q ) is given by thenumber of elements ε in H ∗ ( Q ) with λ < ε ≤ λ . By (6) H ∗ ( Q ) containsevery integer between λ and λ , meaning that the codimension is exactly λ − λ = j − i + 1 .To prove the first equalities in (19) and (20), we use Proposition 3 toobtain σ ( λ ) = q − λ and µ ( λ ) = ( i + 1)( j + 1) . Applying (4) and (5)then implies that the relative distances are at least q − λ and ( i + 1)( j + 1) ,respectively. That these are indeed equalities follows from the observationsfollowing Proposition 3.In (19) the last equality follows from the last part of Proposition 3. For(20) the observations in [8, Rem. 4] imply that (3) is in fact an equality.Thus, the minimal distance of d (cid:0) C L ( D, λ Q ) ⊥ (cid:1) is given by µ (( i + j )( q +1)) = i + j + 1 if i + j < q and µ (cid:0) ( i − ( q − − j )) q + ( q − q + 1) (cid:1) = q ( i + j − q + 2) ( i + 1)( j + 1) occurs if and only if i = 0 , and in the second if and only if j = q − .In the above construction we only consider values of i less than q . A similartechnique can be used for q − q ≤ i < q . We state the proposition, butomit the proof since it follows by similar arguments as above. Proposition 21.
Let λ = ( q − − i ) q + ( q − − j )( q + 1) ∈ H ∗ ( Q ) where i ≤ j < q , and define λ = ( q − − j ) q + ( q − − i )( q + 1) − . Then C L ( D, λ Q ) (cid:40) C L ( D, λ Q ) have codimension (cid:96) = j − i + 1 , and their relativedistances satisfy d (cid:0) C L ( D, λ Q ) , C L ( D, λ Q ) (cid:1) = ( i + 1)( j + 1) ≥ d ( C L ( D, λ Q )) , (21) and d (cid:0) C L ( D, λ Q ) ⊥ , C L ( D, λ Q ) ⊥ (cid:1) = q − iq − j ( q + 1) = d (cid:0) C L ( D, λ Q ) ⊥ (cid:1) . The inequality in (21) is strict if i (cid:54) = 0 and j (cid:54) = q − . By applying one of Theorems 1 and 2, we can transform Propositions 20and 21 into information on improved linear ramp secret sharing schemesand improved asymmetric quantum codes, respectively. The details of thistranslation are left for the reader.
Having presented two improved constructions of nested code pairs in Sec-tions 5 and 6, this section is devoted to the comparison between the corre-sponding asymmetric quantum codes and codes that already exist in theliterature. The codes are also compared with the Gilbert-Varshamov boundfor asymmetric quantum codes. Moreover, we compare the correspondingsecret sharing schemes with a recent lower bound on the threshold gap [3].When presenting code parameters we give the actual codimension ratherthan using the estimates in Section 5 which rely on the bounds in Proposi-tions 7 and 8.Since the codes obtained in Sections 5 and 6 are relatively long comparedto the field size, the literature does not contain many immediately compar-able codes. Yet, one way to obtain such codes is by using Construction IIof La Guardia [17, Thm. 7.1], which gives asymmetric quantum generalizedReed-Solomon codes. Adjusting the theorem to codes over F q gives thefollowing result. 18 heorem 22. Let q be a prime power. There exist asymmetric quantumgeneralized Reed-Solomon codes with parameters [[ m m , m (2 k − m + c ) , ≥ d/ ≥ d − c ]] q where < k < m < k + c ≤ q m and k = m − d + 1 , and where m , d > c + 1 , c ≥ , and m ≥ are integers. Example 2.
By using different values for the parameters in Theorem 22,we obtain asymmetric quantum codes of varying lengths. If the chosenparameters give a code of length less than q , we can pad each codeword withzeroes in order to obtain the correct length. Note that this does not changethe relative distance of the nested codes nor of their duals.For q = 3 Table 1 lists the best code parameters that can be obtainedin this way together with the comparable codes from the constructions inSections 5 and 6. In the third column, the parameter d z is maximized underthe condition that the dimension and the distance d x are at least as high asin [17]. In the fourth, the dimension is maximized, keeping at least the sameminimal distances. As is evident, the codes of the present paper performvery favourably.We further note that all presented new codes exceed the Gilbert-Varshamovbound for asymmetric quantum codes [18, Thm. 4]. Additionally, we remarkthat nesting usual one-point Hermitian codes and using the Goppa bounddoes not provide asymmetric quantum codes as good as the ones in columnsthree and four. The two constructions in Theorem 24 and Corollary 29 of [7] based on codesdefined from Cartesian product point sets provide another way to obtainasymmetric quantum codes that can be compared to the ones in this paper.We summarize these constructions in the following two theorems.
Theorem 23.
Consider integers m ≥ and s ≤ q where q is a primepower. Given δ ∈ { , , . . . , s m } define v ∈ { , , . . . , m − } such that s v ≤ δ ≤ s v +1 , and choose an integer δ ≤ (cid:98) ( s − δ /s v + 1) s m − v +1 (cid:99) . Thenthere exists an asymmetric quantum code with parameters [[ s m , (cid:96), δ /δ ]] q where (cid:96) ≥ s m − m (cid:88) t =1 t − (cid:32) δ (cid:18) ln (cid:18) s m δ (cid:19)(cid:19) t − + δ (cid:18) ln (cid:18) s m δ (cid:19)(cid:19) t − (cid:33) . heorem 24. Consider integers < s ≤ q where q is a prime power, andlet m ∈ { , , . . . , s − } . Then for any (cid:96) ≤ m + 1 such that (cid:96) is even if andonly if m is odd, there exists an asymmetric quantum code with parameters [[ s , (cid:96), d z /d x ]] q where the distances are d z = (cid:0) s − ( m − (cid:96) + 1) (cid:1)(cid:0) s − ( m + (cid:96) − (cid:1) and d x = ( m − (cid:96) + 3)( m + (cid:96) + 1) . The two distances may also be interchanged. Construction of [17, Thm. 7.1] This paper ( m , m , k, c ) Code d z maximized (cid:96) maximized (2 , , ,
10) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
8) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
6) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
4) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
2) [[27 , , / [[27 , , / [[27 , , / (3 , , ,
4) [[27 , , / [[27 , , / [[27 , , / (3 , , ,
2) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
9) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
7) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
5) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
3) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
1) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
8) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
6) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
4) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
2) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
7) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
5) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
3) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
1) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
6) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
4) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
2) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
5) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
3) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
1) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
4) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
2) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
3) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
1) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
2) [[27 , , / [[27 , , / [[27 , , / (2 , , ,
1) [[27 , , / [[27 , , / [[27 , , / Table 1.
Asymmetric quantum codes of length over F . The first column states theparameters used in Theorem 22 to obtain the codes in the second. If necessary, thesehave been padded with zeroes to obtain length . The codes in the third and fourthcolumns are based on the construction in Sections 5 and 6. Example 3.
By using different parameters in Theorems 23 and 24 andpadding with zeroes if necessary, we obtain the asymmetric quantum codes resented in Table 2. This table also shows comparable codes from theconstructions in Sections 5 and 6 which have either d z or (cid:96) maximized as inExample 2. From the table it is evident that the codes of the present paperperform very favourably.Again, we further note that all presented new codes exceed the Gilbert-Varshamov bound [18, Thm. 4], and that these codes cannot be constructedusing information from the Goppa bound applied to nested one-point Hermi-tian codes. Example 4.
A few codes of length over F are given in [6]. The construc-tion in Section 5 can match – but not improve on – the codes [[8 , , / , [[8 , , / , [[8 , , / , [[8 , , / , and [[8 , , / . Additionally, [6]presents a code with parameters [[8 , , / , where we can construct an [[8 , , / -code instead. All of these codes exceed the quantum Gilbert-Varshamov bound [18, Thm. 4], and the Goppa bound applied to nestedone-point Hermitian codes cannot provide such parameters.Some codes over F are presented as well. These codes do, however, havea length that is at least . When presenting constructions of codes, it is customary to compare it totables of ‘best known’ linear codes such as [20, 11]. Unfortunately, similartables do not exist for asymmetric quantum codes. As we shall recall in amoment, however, one can still measure asymmetric quantum codes againstthe usual bounds on linear codes.Before doing so, we observe that the tables in [11] only contains alphabetsup to F , whereas [20] has F as its largest alphabet. As indicated by thefollowing example, however, the latter tables are generally not as optimizedas the ones by [11]. Example 5.
In some cases the improved codes (cid:101) E ( δ ) exceed the codes givenby [20]. For instance, when considering codes over F , the codes (cid:101) E (12) , (cid:101) E (9) , and (cid:101) E (8) with parameters [64 , , , [64 , , , and [64 , , ,respectively, all provide a minimal distance that is one higher than thecorresponding code in the table.Over F the same is true for the codes (cid:101) E (20) , (cid:101) E (16) , and (cid:101) E (12) , whichhave parameters [125 , , , [125 , , , and [125 , , . Addi-tionally, (cid:101) E (15) has parameters [125 , , , exceeding the table distanceby . Recall from Section 6 that our second construction of nested code pairs(which are code pairs of small codimension) gives impure asymmetric quantumcodes. This is already an advantage as the error-correcting algorithms can21ake advantage of the impurity. Another advantage of considering relativedistances rather than only minimal distances emerges when analysing theerror-correcting ability of asymmetric quantum codes. In order to illustratethis advantage, we can compare nested codes from the construction in Sec-tion 6 with pairs of best known linear codes from the tables in [11, 20].Note that the pairs of best known linear codes from such tables generally donot result in nested code pairs; that is, they are not guaranteed to satisfythe requirement that the dual of one code is contained in the other. Thecomparison with tables of best known linear codes is done in the followingexample. Whenever the tables of [20] are considered, we will use the min-imum distance of an improved algebraic geometric Goppa code from theHermitian curve if this exceeds the table value as in Example 5.
Example 6.
Having fixed a code pair C ⊂ C ⊆ F nq of codimension (cid:96) and d ( C ) = δ , we consider the greatest value g ( (cid:96), δ ) such that the tables of bestknown linear codes ensure the existence of C, C (cid:48) ⊆ F nq with dim C − dim C (cid:48) = (cid:96) , d ( C ) = δ , and d ( C (cid:48)⊥ ) ≥ g ( (cid:96), δ ) . This is the same method as used in[7], and bears resemblance to the idea in [6, Thm. 2]. Using this procedureit is in no way guaranteed that C (cid:48) ⊂ C . However, as shown in Table 3 theconstruction in Section 6 is in many cases on par with the best known codes,while simultaneously guaranteeing the inclusion C ⊂ C . In some cases theuse of relative distances will even exceed the values obtained from the bestknown codes. As in the previous examples, the codes in Table 3 all exceedthe Gilbert-Varshamov bound for asymmetric quantum codes [18, Thm. 4]. Turning to secret sharing schemes, [3] presents a lower bound on thethreshold gap r − t . That is, the authors bound the smallest possibledifference between the reconstruction number r and the privacy number t for q -ary linear ramp secret sharing schemes with n shares and secrets fromfrom F (cid:96)q . For linear ramp secret sharing schemes over F q , they show that r ≥ t + ( q m − n + 2) + ( q m +2 − q m )( (cid:96) − m ) q m +2 − (22)for every m ∈ { , , . . . , (cid:96) − } . Of course, one should choose the m thatgives the best bound. Comparing the secret sharing schemes obtained in thispaper with the bound (22) helps to quantify how optimal the construction is.This is done in the following example, which also illustrates the advantageof using the improved codes from Section 5 and the improved informationfrom Section 6 rather than relying solely on the Goppa bound applied tonested one-point Hermitian codes. 22 onstructions of [7] This paperType ( s, m ) Code d z maximized (cid:96) maximized Thm. 24 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 24 (5 ,
4) [[27 , , / [[27 , , / [[27 , , / Thm. 24 (5 ,
1) [[27 , , / [[27 , , / [[27 , , / Thm. 24 (5 ,
3) [[27 , , / [[27 , , / [[27 , , / Thm. 24 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 24 (5 ,
4) [[27 , , / [[27 , , / [[27 , , / Thm. 24 (5 ,
3) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 24 (5 ,
4) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Thm. 23 (5 ,
2) [[27 , , / [[27 , , / [[27 , , / Table 2.
Asymmetric quantum codes of length over F . The first column indicateswhether Theorem 23 or 24 was used in the codes given in the third column. The secondstates the parameters used, except for those that can be read off directly from the code.The codes in the fourth and fifth columns are based on the construction in Sections 5and 6. ( i, j ) Parameters g ( (cid:96), δ ) (2 ,
2) [[27 , , / ,
1) [[27 , , / ,
2) [[27 , , / ,
1) [[27 , , / ,
2) [[27 , , / ,
3) [[64 , , / , [[64 , , / ,
1) [[64 , , / ,
3) [[64 , , / ∗ (1 ,
2) [[64 , , / ,
1) [[64 , , / ,
3) [[64 , , / ∗ (0 ,
2) [[64 , , / ,
3) [[64 , , / ( i, j ) Parameters g ( (cid:96), δ ) (4 ,
4) [[125 , , / , [[125 , , / ,
2) [[125 , , / – (1 ,
1) [[125 , , / – (3 , [[125 , , / , [[125 , , / ,
2) [[125 , , / – (0 ,
1) [[125 , , / – (2 , [[125 , , / ,
3) [[125 , , / – (0 ,
2) [[125 , , / – (1 ,
4) [[125 , , / ,
3) [[125 , , / – (0 ,
4) [[125 , , / Table 3.
Comparing the asymmetric quantum codes from Section 6 with the best knowncodes. For q = 3 [11] is used, and for the remaining values of q , [20] is used. The codesmarked in bold exceed g ( (cid:96), δ ) , and the values of g ( (cid:96), δ ) marked with an asterisk stemfrom the improvements in Example 5. A dash indicates that the tables do not containenough information to determine g ( (cid:96), δ ) . ‘r Reconstruction number for q = 3 , t ≥ ‘r Reconstruction number for q = 4 , t ≥ ‘r Reconstruction number for q = 5 , t ≥ ‘r Reconstruction number for q = 7 , t ≥ Figure 1.
The minimal achievable reconstruction number r given a desired privacynumber t . The plots show the constructions from Sections 5 and 6, a construction usingthe Goppa bound only, and the lower bound from (22). Example 7.
In each of the plots in Figure 1, a desired privacy number t has been fixed. For each codimension the plots then show the minimalreconstruction number r achievable with the constructions from Sections 5and 6 when privacy number at least t is required. The plots also show thelower bound in (22) .Recall that the codes under consideration have length q , meaning thatthe four corresponding secret sharing schemes support , , , and participants, respectively. As the plots demonstrate, the constructions ofthis paper provide secret sharing schemes with parameters that could not beobtained by using nested Hermitian one-point codes and the Goppa boundalone.We remark that the four given examples use a relatively large privacy arameter t . Yet, it is also possible to obtain improved reconstruction num-bers for small values of t . In this paper we presented two improved constructions of nested code pairsfrom the Hermitian curve, and gave a detailed analysis of their performancewhen applied to the concepts of secret sharing and asymmetric quantumcodes. Regarding information leakage in secret sharing we studied the recon-struction number r and the privacy number t , which give information on fullrecovery and full privacy, respectively. We note that it is possible to obtaininformation about partial information leakage by studying relative general-ized Hamming weights rather than just relative minimum distances [15, 9].For asymmetric quantum codes we applied the CSS construction. Applyingthe method of Steane’s enlargements [22] is a future research agenda. Acknowledgements
The authors would like to thank Ignacio Cascudo for helpful discussions.
Appendix A Additional results on σ and µ In this section we state a number of lemmas that are needed in Sections 3and 4. The lemmas all follow as corollaries to Proposition 3. To aid thereader in understanding them more easily, we first give an example forreference.
Example 8.
In Table 4 we list H ∗ ( Q ) , σ ( H ∗ ( Q )) , and µ ( H ∗ ( Q )) for theHermitian function field over F , i.e. for q = 4 . Entries are orderedaccording to ( i, j ) where λ = iq + j ( q + 1) . The first lemma explains when (2) equals the Goppa bound and when it issharper.
Lemma 25.
For all λ = iq + j ( q + 1) ∈ H ∗ ( Q ) it holds that σ ( λ ) ≥ n − λ where n = q . The inequality is strict if and only if q − q ≤ i < q and ≤ j < q } . The next five lemmas give information on the relation between the val-ues σ ( iq + j ( q + 1)) and σ ( i (cid:48) q + j (cid:48) ( q + 1)) for different constellations of i, j, i (cid:48) , j (cid:48) . Using the translation from σ to µ as given in Proposition 3, thissimultaneously implies relations on µ .25 Table 4.
Upper table: H ∗ ( Q ) . Middle table: σ ( H ∗ ( Q )) . Lower table: µ ( H ∗ ( Q )) Lemma 26.
For < i ≤ q − q − and ≤ j < q − it holds that σ ( iq + j ( q + 1)) = σ (( i − q + ( j + 1)( q + 1)) + 1 . Furthermore, for ≤ i ≤ q − q − it holds that σ ( iq + ( q − q + 1)) = σ (( i + q ) q ) + 1 . Lemma 27.
The sequence (cid:0) σ (0 · q ) , σ ( q ) , . . . , σ (( q − q ) , σ (( q − q + ( q + 1)) ,σ (( q − q + 2( q + 1)) , . . . , σ (( q − q + ( q − q + 1)) (cid:1) is strictly decreasing. Lemma 28.
We have σ (( q − q + s ) q + t ( q + 1)) = σ (( q − q + t ) q + s ( q + 1)) for ≤ s, t < q − . Lemma 29.
Given q − q ≤ i ≤ q − then for non-negative s such that q − q ≤ i − s we have σ (( i − s ) q + s ( q + 1)) ≥ σ ( iq ) . Similarly, given ≤ j ≤ q − then for non-negative s such that j + s ≤ q − we have σ (( q − − s ) q + ( j + s )( q + 1)) ≥ σ (( q − q + j ( q + 1)) . Lemma 30. If σ ( iq + j ( q + 1)) ≤ q then q − q ≤ i < q . Finally, we present a lemma on the relation between σ ( λ ) and µ ( λ ) for λ belonging to a certain window. Lemma 31.
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