Codes with locality from cyclic extensions of Deligne-Lusztig curves
CCodes with locality from cyclic extensions ofDeligne-Lusztig curves
Gretchen L. Matthews and Fernando Pi˜nero Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 [email protected] (cid:63)
Department of Mathematics, University of Puerto Rico at Ponce, Ponce, PR [email protected]
Abstract.
Recently, Skabelund defined new maximal curves which arecyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve.In this paper, we consider locally recoverable codes constructed fromthese new curves, complementing that done for the GK curve. Locallyrecoverable codes allow for the recovery of a single symbol by accessingonly a few others which form what is known as a recovery set. If everysymbol has at least two disjoint recovery sets, the code is said to haveavailability. Three constructions are described, as each best fits a partic-ular situation. The first employs the original construction of locally re-coverable codes from curves by Tamo and Barg. The second yields codeswith availability by appealing to the use of fiber products as describedby Haymaker, Malmskog, and Matthews, while the third accomplishesavailability by taking products of codes themselves. We see that cyclicextensions of the Deligne-Lusztig curves provide codes with smaller lo-cality than those typically found in the literature.
Maximal curves have played a role in a number of applications in coding theory.For instance, they allow for the construction of long algebraic geometry codes andyield explicit families of codes with parameters exceeding the Gilbert-Varshamovbound [23]. The Deligne-Lusztig curves, which include the Hermitian, Suzuki,and Ree curves, have proven particularly useful. In particular, Hermitian codesare perhaps the best understood algebraic geometry codes other than Reed-Solomon codes. The Suzuki and Ree curves share several important propertieswith the Hermitian family in that they are optimal with respect to the Hasse-Weil bound and have known automorphism groups; thus, codes from these curveshave interesting properties as well.More recently, maximal curves have been employed in the construction ofcodes with locality. In some applications, it is desirable to recover a single (or (cid:63)
Partially supported by NSF DMS-1855136. a r X i v : . [ c s . I T ] J un G. L. Matthews and F. Pi˜nero small number of) codeword symbol(s) by accessing only a few, say r , particularsymbols of the received word. This leads to the notion of locally recoverablecodes, or LRCs. Tamo and Barg [21] introduced a construction for codes withlocality that is similar to that of algebraic geometry codes. This motivated muchwork on locally recoverable codes, including [1], [2], [7], [11], [13]. In [9], we em-ploy maximal curves to construct LRCs with availability t ≥
2, meaning eachcoordinate j has t disjoint recovery sets. Codes with availability make informa-tion more available to more users, since recovery of an erasure is not entirelydependent on a single set of coordinates (which may itself contain erasures).In this paper, we define codes with locality from new maximal curves con-structed by Skabelund [18] using cyclic covers of the Suzuki and Ree curves. TheSuzuki curve S q over F q gets its name from its automorphism group which is theSuzuki group Sz ( q ) of order q ( q + 1)( q − F q whose automorphismgroup is Sz ( q ). Skabelund considers a cyclic extension of S q and proves it ismaximal over F q and F q . Similarly, the Ree curve R q over F q has a Ree groupas its automorphism group. Both curve constructions are similar to that of theGiulietti-Korchm´aros, or GK, curve, which has already proven useful in con-structing codes with locality. These cyclic extensions of the Suzuki and Reecurves have also been utilized for AG codes and for quantum codes from them[16] and their automorphism groups have been determined by Giulietti, Mon-tanucci, Quoos, and Zini [6].This paper is organized as follows. In Section 2, we obtain codes with localityfrom the cyclic extension ˜ S q of the Suzuki curve S q and the cyclic extension ˜ R q ofthe Ree curve R q . The locality is much smaller relative to the alphabet size andcode length than comparable constructions. In Section 3, we construct codeswith availability from ˜ S q and ˜ R q . Our constructions build on tools found in[21] and [9], and some useful background may be found there. Because explicitcode descriptions remain out of reach for these standard constructions whenemploying R q or ˜ R q (as they depend on explicit bases for Riemann-Roch spaceswhich remain elusive), we provide an alternate construction for such settings.We also consider constructions from products of codes. In Section 4, we considerexamples of the above constructions and make some comparisons between them. Locally recoverable codes, or LRCs for short, can recover a single (or smallnumber of) codeword symbol(s) by accessing a small number, say r , of particularsymbols of the received word. In principle, the locality r should be small so asto limit network traffic though this can adversely impact other code parameters.While an [ n, k, d ] code C , meaning a code of length n , dimension k , and minimumdistance d , can recover any d − (cid:4) d − (cid:5) errors, this assumesaccess to all other symbols of the entire received word. More precisely, the code odes with locality from cyclic extensions of Deligne-Lusztig curves 3 C of length n over the alphabet F (typically taken to be a finite field) is locallyrecoverable with locality r if and only if for all j ∈ [ n ] := { , . . . , n } there exists A j ⊆ [ n ] \ { j } with | A j | = r and c j = φ j ( c | A j )for some function φ j : A j → F for all c ∈ C . The set A j is called a recovery set for the j -th coordinate. In thissection, we see how cyclic extensions naturally lead to LRCs. The Suzuki curve S q may be described by the equation S q : y q + y = x q ( x q + x )where q = 2 s , q = 2 q , and s ∈ N . It is an optimal curve over F q , having q + 1 F q -rational points. Indeed, if a, b ∈ F q , a q = a and b q = b ; since char F q = 2, b q + b = 0 = a q ( a q + a ). In addition, there is a unique point at infinity P ∞ corresponding to x = z = 0 and y = 1. The genus of S q is q ( q −
1) [8, Lemma1.9]. It is maximal over F q , having q + 1 + 2 q q ( q − F q -rational points [4,Equation (7)]. Define ˜ S q : (cid:40) y q + y = x q ( x q + x ) t m = x q + x. where m = q − q + 1. The curve ˜ S q has a unique point at infinity, and affinepoints will be denoted P abc := ( a : b : c : 1) to mean the unique zero of x − a , y − b , and t − c , just as those of S q will be denoted by P ab . The genus of ˜ S q is q − q + q [18]. According to [18], the number of F q -rational points on ˜ S q thatare not F q -rational is q − q + q − q ;see also [18, Section 3] for a discussion of the points on this curve. Define g : ˜ S q → S q P abc (cid:55)→ P ab Let S := S q (cid:0) F q (cid:1) \ S q ( F q ) . (1)Then | S | = q +2 q q ( q − − q [4, Equations (4)-(7)]. Set D := (cid:80) P ∈D P where D := g − ( S ) = (cid:110) P abc ∈ ˜ S q (cid:0) F q (cid:1) : c (cid:54) = 0 (cid:111) . (2) G. L. Matthews and F. Pi˜nero
For each P ab ∈ S , g − ( P ab ) = { P abc : c m = a q + a } , so | g − ( P ab ) | = q − q + 1 . (3)Recall that given a divisor G on a curve X over a field F , the space offunctions determined by G , sometimes called the Riemann-Roch space of G , is L ( G ) := { f ∈ F ( X ) : ( f ) ≥ − G } ∪ { } , where F ( X ) denotes the set of rational functions on X , and ( f ) denotes thedivisor of the function f ; to say that ( f ) = (cid:80) Q ∈Z a Q Q − (cid:80) P ∈P b P P with a Q , b P ∈ Z + means f has a zero of order a Q at Q and a pole of order b P at P . We use the standard notation ( f ) := (cid:80) Q ∈Z a Q Q to denote the zero divisorof f and ( f ) ∞ := (cid:80) P ∈P b P P to denote the pole divisor of f . Let α ∈ Z + , andconsider the divisor G := α P ∞ + (cid:88) a,b ∈ F q P ab on S q . It is worth noting that L (cid:16) α (cid:16) P ∞ + (cid:80) a,b ∈ F q P ab (cid:17)(cid:17) ∼ = L (cid:0) α (cid:0) q + 1 (cid:1) P ∞ (cid:1) [4]. According to [4, Theorem 1], a basis for L ( G ) is given by B := x a y b u c v d ( x q + x ) e : aq + b ( q + q ) + c ( q + 2 q )+ d ( q + 2 q + 1) ≤ α + eq a ∈ { , . . . , q − } , b ∈ { , } ,c, d ∈ { , . . . , q − } , e ∈ { , . . . , α } ⊆ F q ( S q )where u = x q +1 − y q and v = xy q − u q . Set V := (cid:10) f t i : i = 0 , . . . , m − f ∈ B (cid:11) F q . Now define ev : V → F ( q − q +1) ( q +2 q q ( q − − q ) q f (cid:55)→ ( f ( P abc )) P abc ∈ ˜ S q ( F q ) \ ˜ S q ( F q ) , and set C ( D, G, g ) := ev ( V ). Note that the evaluation map ev is well-defined,as | D | = ( q − q + 1)( q + 2 q q ( q − − q ) and f ∈ V has no poles at pointsin D . One may notice that V ⊆ L (cid:0) mα + ( m − q (cid:1) ˜ P ∞ + mα (cid:88) a,b ∈ F q P a,b, (4) odes with locality from cyclic extensions of Deligne-Lusztig curves 5 where ˜ P ∞ denotes the unique point of ˜ S q lying above P ∞ . Let G (cid:48) := (cid:0) mα + ( m − q (cid:1) ˜ P ∞ + mα (cid:88) a,b ∈ F q P a,b, . We are now ready to state the result.
Theorem 1.
Suppose C ( D, G, g ) is constructed as above where deg G (cid:48) < | S | .Then C ( D, G, g ) is an [ n, k, d ] code over F q with locality q − q , n = ( q − q + 1) (cid:0) q + 2 q q ( q − − q (cid:1) ,k = ( q − q ) (cid:0) α (cid:0) q + 1 (cid:1) − q ( q −
1) + 1 (cid:1) , and d ≥ n − (cid:0) mαq + mα + ( m − q (cid:1) . Proof.
The map ev is injective, since deg G (cid:48) < | S | guarantees the kernel of theevaluation map is { } ; this may be observed by noting that if f ∈ ker ev \ { } then f would have more zeros than poles. Hence, the dimension is given bydim F q V which follows from the facts that (cid:8) t i : i = 0 , , . . . , m − (cid:9) is a basis of F q ( ˜ S q ) / F q ( S q ); B is a basis for F q ( S q ) / F q ; and | B | = (cid:0) α (cid:0) q + 1 (cid:1) − q ( q −
1) + 1 (cid:1) according to [4, Remark 1]. We claim that R := g − ( P ab ) \ { P abc } is a recoveryset for the position corresponding to P abc . Suppose f ∈ V . Then f ( x, y, t ) = m − (cid:88) i =0 M (cid:88) j =1 a ij f ∗ j t i for some a ij ∈ F q and f ∗ j ∈ B , where M := | B | . Notice that f ( a, b, T ) ∈ F q [ T ]and deg T f ( a, b, T ) ≤ m −
2. Hence, f ( a, b, c ) can be recovered using the m − P abc (cid:48) ∈ R . As a result, f ( P abc ) may be recovered using onlyelements of R .To determine a bound on the minimum distance d , we use that d ≥ wt ( ev ( h )) ≥ n − deg( h ) where h = f t m − and f ∈ B ⊆ L ( G ). Then deg( h ) ≥ m deg( G ) + ( m − q ≥ mα ( q + 1) + ( m − q as G is a divisor of degree α ( q + 1) on S q , [ ˜ S q : S q ] = m ,and ( t ) is a divisor on ˜ S q with zero divisor of degree q . As a result d ≥ n − (cid:0) mα ( q + 1) + ( m − q (cid:1) . Alternatively, the bound on the minimum distance may be seen as a consequenceof d ≥ n − deg G (cid:48) using (4). G. L. Matthews and F. Pi˜nero
Example 1.
Let q = 8 and q = 2, so q = 4096. Notice that the Suzuki curve S : y + y = x (cid:0) x + x (cid:1) has 64 F -rational points and 5888 F -rational points. Here, | S | = 5824 and n = 29120. Then C ( D, G, g ) has locality 4. We can compare this with an LRC C (cid:48) from the Hermitian curve y + y = x over the same field, F . Usinga projection onto the x -coordinate gives a code of length 262144 with locality63 whereas projection onto the y -coordinate yields locality 64. Hence, the con-struction using ˜ S has a smaller ratios of locality to code length and to alphabetsize. Remark 1.
1. Other bounds on the minimum distance of the codes in Theorem1 may be given; see [22] for instance.2. Alternatively, an LRC may be constructed using the projection g : ˜ S q → C m P abc (cid:55)→ Q ac where C m denotes the curve given by t m = x q + x and Q ac denotes thecommon zero of x − a and t − c . Let S be as in (1), D as in (2), and G (cid:48) := αQ ∞ where Q ∞ is the point at infinity on C m . Then a basis for L ( αQ ∞ ) is givenby B (cid:48) := (cid:8) t i x j : i ≥ , j ∈ { , . . . , q − } , qi + mj ≤ α (cid:9) ;see, for instance, [10, Lemma 12.2(i)]. Use this to define V = (cid:10) f y i : i ∈ { , . . . , q − } , f ∈ B (cid:48) (cid:11) . The code C ( D, G (cid:48) , g ) has locality q − q − |B| .In Section 3, we will see how these two approaches can be combined to giveLRCs with availability. Before doing so, we turn our attention to cyclic extensionsof Ree curves. The Ree curve R q may be described by the equation R q : (cid:40) y q − y = x q ( x q − x ) z q − z = x q ( x q − x )where q = 3 s , q = 3 q , and s ∈ N . It is optimal over F q . In addition, there is aunique point at infinity. The genus of R q is q ( q −
1) ( q + q + 1) [8]. Define˜ R q : y q − y = x q ( x q − x ) z q − z = x q ( x q − x ) t m = x q − x odes with locality from cyclic extensions of Deligne-Lusztig curves 7 where m = q − q + 1. The curve ˜ R q has a unique point at infinity, and affinepoints will be denoted P abcd := ( a : b : c : d : 1) to mean the unique zero of x − a , y − b , z − c and t − d , just as those of R q will be denoted by P abc . The genus of˜ R q is q − q + q . According to [16], the number of F q -rational points on ˜ R q thatare not F q -rational is q − q + q − q ;see also [18]. Define g : ˜ R q → R q P abcd (cid:55)→ P abc and let S := R q (cid:0) F q (cid:1) \ R q ( F q ) . (5)Set D := (cid:80) P ∈D P where D := g − ( S ) = (cid:110) P abcd ∈ ˜ R q (cid:0) F q (cid:1) : d (cid:54) = 0 (cid:111) . (6)For each P abc ∈ S , g − ( P abc ) = { P abcd : d m = a q − a } , so | g − ( P abc ) | = q − q + 1 . (7)Consider the divisor G = αP ∞ on R q with m deg G + ( m −
2) deg( t ) ∞ < | S | .Set V := (cid:10) f t i : i = 0 , . . . , m − f ∈ L ( G ) (cid:11) F q . Now define ev : V → F |D| q f (cid:55)→ ( f ( P abcd )) P abcd ∈ ˜ R q ( F q ) \ ˜ R q ( F q ) , and set C ( D, G, g ) := ev ( V ). Proposition 1.
Suppose C ( D, G, g ) is constructed as above. Then C ( D, G, g ) is an [ q − q + q − q , ( m − (cid:96) ( G )] code over F q with locality q − q .Proof. This follows similarly to that of Theorem 1.
Remark 2.
1. Explicit bases for L ( G ) where G is a divisor on the Ree curveis a topic of current research for arbitrary q , even for the case where G is a multiple of the point at infinity. See [19] for recent work on relatedtopics. The work [3] also highlights the challenges of this problem, whichwas originally stated in [17]; indeed, when s = 1 (so q = 27), the associatedWeierstrass semigroup has more than 100 generators, compared with 2 in theHermitian case and 4 for Suzuki. Hence, the dimension of the codes describedin Proposition 1 cannot be specified more precisely the expression givenabove for arbitrary q . However, for specific small values of q , a set of functionswhich generate L ( G ) may be found computationally. We include this resultso that if the theory progresses and sheds more light on this value, LRCsare an immediate consequence. We also note that our interest in the Ree G. L. Matthews and F. Pi˜nero curve is partially motivated by the fact that it allows for results over fieldsof odd characteristic whose cardinalities are odd powers of primes (unlikethe Suzuki curve, which is considered over a field of even characteristic, andthe Hermitian curve which is considered over a field with square cardinality).2. Also, as in Remark 1, the projection g : ˜ R q → C m P abcd (cid:55)→ Q ad , where C m : t m = x q − x and Q ad denotes the common zero of x − a and t − c , may be used to define a code with different recovery sets than thoseconsidered above.3. A bound on the minimum distance is given in [9, Theorem 3.1] If every coordinate j has t disjoint recovery sets, then C is said to have availability t to reflect that information is more available to users in the presence of erasure.In [9], fiber products of curves are used to construct locally recoverable codeswith availability. We review the construction in the case t = 2 below.Suppose X = Y × Y Y where Y , Y , and Y are curves over a finite field F with rational, separable maps h i : Y i → Y . The F q -rational points of X are { ( P , P ) : P i is an F q − rational point on Y i , h ( P ) = h ( P ) } . Thus, there areprojection maps g i : X → Y i defined by g i ( P , P ) = P i ; a rational, separa-ble map g : X → Y given by g = h ◦ g = h ◦ g ; maps of function fields h ∗ i : F ( Y ) → F ( Y ) given by h ∗ i ( f ) := f ◦ h i ; and primitive elements x i of theextensions F ( Y i ) /h ∗ i ( F ( Y )) . Let S be a set of F -rational points on Y , and take D := (cid:80) P ∈ g − ( S ) P . Choose an effective divisor G on Y of degree (cid:96) < | S | , andtake a basis { f , . . . , f t } for L ( G ). Set V := Span { ( f i ◦ g ) x ∗ e x ∗ e : 1 ≤ i ≤ t, ≤ e i ≤ deg h i − } where x ∗ i = g ∗ i ( x i ) given that g ∗ i : F ( Y i ) → F ( X ) for i = 1 ,
2. Consider ev : V → F n f (cid:55)→ ( f ( P i )) P i ∈ supp D . Then the code C ( D, G, g, g , g ) := ev ( V ) has length | D | = deg g | S | , dimension t (deg h −
1) (deg h − , and minimum distance bounded below according to [9]. For i = 1 , g − i ( g i ( Q )) \ { Q } serves as a recovery set for Q ∈ S . Hence, C ( D, G, g, g , g ) has locality 2. Nextwe apply this construction to ˜ S q and ˜ R q . odes with locality from cyclic extensions of Deligne-Lusztig curves 9 Cyclic extensions of Suzuki curves as fiber products.
Because ˜ S q is thefiber product of covers S q → P x and C m → P x , we may apply the constructionto obtain a code with availability 2 and localities m − q −
1; that is,every coordinate has 2 disjoint recovery sets, one of cardinality q − q and oneof cardinality q −
1. To do this, consider the projection maps g : ˜ S q → C m , g : ˜ S q → S q , and g : ˜ S q → P x . We take S as in (1), D as in (2), and G := αP ∞ where P ∞ is the unique point at infinity on P x . Fix a basis B of L ( G ), and V := (cid:10) f y i t j : 0 ≤ i ≤ q − , ≤ j ≤ m − , f ∈ B (cid:11) F q . P x S q C m ˜ S q h h g g g Fig. 1.
Cyclic extension of Suzuki curve viewed as a fiber product
Theorem 2.
Suppose C ( D, G, g, g , g ) is constructed as above. Then the code C ( D, G, g, g , g ) is an [ n, k, d ] code over F q with availability and recovery setsfor each coordinate of sizes q − q and q − , where n = ( q − q + 1) (cid:0) q + 2 q q ( q − − q (cid:1) ,k = ( q − q ) ( α + 1) ( q − , and d ≥ n − (cid:0) αmq + ( q − m ( q + q ) + ( m − q (cid:1) . Proof.
The length and dimension can be verified directly by applying [9, The-orem 3.1]. To determine the minimum distance d , we use the fact that d ≥ n − wt ( ev ( h )) ≥ n − deg( h ) where h = f y q − t m − and f ∈ L ( αP ∞ ). Then( h ) = ( f ) + ( q − y ) + ( m − t ). Note that when considered as a functionson ˜ S q , deg( f ) ≤ αmq , deg( y ) ≤ m ( q + q ), and deg( t ) ≤ q . Putting this to-gether, we conclude that d ≥ n − (cid:0) αmq + ( q − m ( q + q ) + ( m − q (cid:1) , whichcoincides with that given in [9, Theorem 3.1].We claim that R (1) := g − ( g ( P abc )) \ { P abc } = (cid:8) P ab (cid:48) c : b (cid:48) ∈ F q \ { b } (cid:9) and R (2) := g − ( g ( P abc )) \ { P abc } = (cid:8) P abc (cid:48) : c (cid:48) ∈ F q \ { c } (cid:9) are recovery sets for the position corresponding to P abc . Suppose f ∈ V . Then f ( x, y, t ) = (cid:80) m − i =0 (cid:80) Mj =1 a ij f ∗ j t i . Notice that f ( a, b, T ) ∈ F q [ T ] and the degreeis bounded by deg T f ( a, b, T ) ≤ m −
2. Hence, f ( a, b, c ) can be recovered usingthe m − P abc (cid:48) ∈ R . As a result, f ( P abc ) may be recoveredusing only elements of R .Observe that the functions in the set V are modified from the constructionin Section 2 in order to obtain multiple recovery sets for each position, thusimpacting the dimension of the code.One might compare this with the code found in [9, Theorem 6.1], which hasavailability 2 with recovery sets of size q −
1, length n = q ( q − q +2 qq + q +1)and dimension k = ( q − q − q + 2 qq + q + 1). Notice that the new codesdefined using ˜ S q give the option of using a smaller recovery set (cardinality q − q compared with q − Cyclic extensions of Ree curves as fiber products.
Because ˜ R q is the fiberproduct of R q → P x and C m → P x , we may apply this construction to obtain acode with availability 2 and localities m − q −
1; that is, every coordinatehas 2 disjoint recovery sets, one of cardinality q − q and one of cardinality q −
1. To do this, consider the projection maps g : ˜ R q → C m , g : ˜ R q → R q ,and g : ˜ R q → P x . We take S as in (5), D as in (6), and G is a divisor on P x . Fixa basis B of L ( G ), and V := (cid:10) f y i t j : 0 ≤ i ≤ q − , ≤ j ≤ q − q − , f ∈ B (cid:11) F q . P x R q C m ˜ R q h h g g g Fig. 2.
Cyclic extension of Ree curve viewed as a fiber product
Proposition 2.
The code C ( D, G, g, g , g ) constructed as above is a code withparameters [ q − q + q − q , (cid:96) ( G )( q − q − q )] code over F q with availability and recovery sets for each coordinate of sizes q − q and q − .Proof. The proof is similar to that of Theorem 2. odes with locality from cyclic extensions of Deligne-Lusztig curves 11
Remark 3.
1. As noted in Remark 2, the explicit construction for codes fromthe Ree curve depends on that of bases for certain Riemann-Roch spaces.We provide an alternate LRC with availability construction from the Reecurve in the Section 4. There, we see codes with more accessible parametersdue to choosing functions to evaluate carefully, rather than beginning withan entire Riemann-Roch space which is difficult to describe.2. A bound on the minimum distance is given in [9, Theorem 3.1].Observe the functions in the set V are modified from the construction in Sec-tion 2 in order to obtain multiple recovery sets for each position, thus impactingthe dimension of the code. We may also take products of codes themselves to obtain LRCs with availability,as detailed below. We begin with the simplest definition, the product of twocodes, C and C , which may be generalized to more factors. Examples of thisconstruction may be found in the next section. Definition 1.
Let C be an [ n , k , d ] code and C be an [ n , k , d ] code overthe same alphabet F . The product code of C and C is defined by assigningsymbols from F to the pairs ( i, j ) ∈ [ n ] × [ n ] such that the symbols assigned in [ n ] × { j } , for j ∈ [ n ] are a codeword in C and { i } × [ n ] for i ∈ [ n ] are acodeword in C ; that is, C × C := { ( a i b j ) ∈ F [ n ] × [ n ] | ( a , a , . . . , a n ) ∈ C , ( b , b , . . . , b n ) ∈ C } An alternative definition is to place symbols from F in an n × n rectangulararray such that each column is a codeword of C and each row is a codeword of C . See also [14]. Theorem 3.
Let C be an [ n , k , d ] code and C be an [ n , k , d ] code. Thenthe code C × C is a [ n n , k k , d d ] code with availability . Moreover, if C has locality r and availability l and C has locality r and availability l , then C × C is a code of availability l + l and locality r + r .Proof. Let D denote the minimum distance of the code C × C . If ( i, j ) is anonzero position, then there are d positions in the set [ n ] × { j } which havea nonzero entry. Suppose those nonzero positions are ( i , j ) , ( i , j ) , . . . , ( i d , j ).For each of those nonzero positions ( i s , j ), there are d nonzero positions in { i s } × [ n ]. Thus there are at least d d nonzero positions.In order to prove equality, let ( a , a , . . . , a n ) be a codeword of weight d in C , and ( b , b , . . . , b n ) be a codeword of weight d in C . Then the codeworddefined by c i,j = a i b j is the required codeword of weight d d .Let I be an information set for C and let I be an information set of C .The i th coordinate of c ∈ C may be written as the linear combination p i m fora message vector m ∈ F I . Likewise, the i th coordinate of c ∈ C may be written as the linear combination q j m for a message vector m ∈ F I . After placingany values in I × I , the remaining values are given by c i,j = p i m q j m .Note that position ( i, j ) is in the two sets [ n ] × { j } and { i } × [ n ]. Thesetwo sets have only ( i, j ) in common. Thus, [ n ] × { j } \ ( i, j ) and { i } × [ n ] \ ( i, j )are recovery sets for ( i, j ); note that they are disjoint as required for availability.Consider ( i, j ) ∈ [ n ] × [ n ]. As C is a code of availability l there are l disjoint sets , I , I , , . . . , I l in [ n ] \ { i } from which position i may be recovered.Likewise as C is a code of availability l there are l disjoint sets , J , J , . . . , J l in [ n ] \{ j } from which position j may be recovered. The sets I ×{ j } , I ×{ j } , . . . , I l × { j } , { i } × J , { i } × J , . . . , { i } × J l are then l + l recovery sets which aredisjoint; this gives the desired availability. A number of examples of LRCs are given in this section, and some comparisonsare drawn between instances of the constructions discussed in this paper as wellas those appearing elsewhere in the literature. In addition, we provide LRCs onthe Ree curve via a construction that allows for computable parameters despitethe issues mentioned in Remarks 2 and 3.Tamo and Barg gave a seminal construction of an optimal LRC code oflocality r in [21]. The LRC construction is based on a set L ⊆ F q , a partition of L into disjoint subsets A , A , ... , A m where each set A i has size r + 1 and apolynomial g ( x ) of degree r + 1 such that g is constant on each subset A i . Tamoand Barg construct an LRC code from a subcode of the Reed–Solomon codeover L of dimension k (cid:48) by evaluating the functions of the form X i g ( X ) j where0 ≤ i ≤ r, i (cid:54) = s for a fixed 0 ≤ s ≤ r and i + ( r + 1) j ≤ k (cid:48) −
1. There are manypartitions and many choices for g ( X ). However, we shall focus on partitionsgiven by linear subsets of F q or by cosets of the multiplicative group of F q . Weshall use evaluation codes as a generalization of Reed–Solomon codes and AGcodes.Let A = { α , α , . . . , α n } ⊆ F mq . Let f ( x , x , . . . , x m ) be a polynomial in m variables. The evaluation map of f on A is defined as ev A : F q [ x , x , . . . , x m ] → F nq where ev A ( f ) = ( f ( α ) , f ( α ) , . . . , f ( α n )) . We remark that the vanishing ideal of A , namely I A = { f ∈ F q [ x , x , . . . , x m ] | f ( α ) = 0 ∀ α ∈ A } , is the kernel of the evaluation map ev A .Let A = { α , α , . . . , α n } ⊆ F mq . Let L be a subspace of F q [ x , x , . . . , x m ].The set C ( A, L ) = { ev A ( f ) | f ∈ L } odes with locality from cyclic extensions of Deligne-Lusztig curves 13 is known as an affine variety code. The definition of an affine variety code simplystates that a linear code may be constructed by evaluating functions on a set ofpoints. In most cases, the structure of L or A will imply certain properties ofthe code hold, such as dimension, minimum distance or locality. Lemma 1.
Let V ⊆ F m q . Let L be a subspace of F q [ x , x , . . . , x m ] . Simi-larly, take V ⊆ F m q and L be a subspace of F q [ y , y , . . . , y m ] . Consider theevaluation codes: C = C ( V , L ) and C = C ( V , L ) . The product code C × C is the evaluation code C = C ( V , L ) , where V = V × V ⊆ F m + m q and theset of evaluated functions is L = { f ( X ) g ( Y ) , f ∈ L , g ∈ L } .Proof. It is clear that taking f ∈ L and g ∈ L and evaluating the product f ( X ) g ( Y ) on the array { ( α, β ) | α ∈ V , β ∈ V } will give a codeword of theform ( f ( α ) g ( β )). This codeword is also a codeword of C × C .In order to prove equality, we use a dimensional analysis. As C is a code ofdimension k , there exist f , f , . . . , f k functions of L and α , α , . . . , α k ∈ V such that f i ( α j ) = δ i,j . Likewise, there exist g , g , . . . , g k functions of L and β , β , . . . , β k ∈ V such that g i (cid:48) ( β j (cid:48) ) = δ i (cid:48) ,j (cid:48) . The evaluation of the functions f i g i (cid:48) on the points α j β j (cid:48) will also imply the image has dimension k k .We will construct LRC codes based on the product code of the Tamo–Bargconstruction and Lemma 1. In particular, we shall take V × V , V × V × V ,for V i ⊆ F q as our evaluation sets and the evaluation functions to be { f ( X ) f ( Y ) f ( Z ) | f i ∈ L i } where L i = { T a g i ( T ) j , ≤ a ≤ r i , a (cid:54) = s i , a + ( r i + 1) j < k (cid:48) } .Note that classical Hermitian codes are obtained by evaluating monomialsof the form M ( s ) := { X i Y j | iq + j ( q + 1) ≤ s } on the q points of the form A = { ( α, β ) ∈ F q | α q +1 = β q + β, α (cid:54) = 0 } . However, as the vanishing ideal of A is the ideal spanned by X q +1 − Y q − Y and X q − −
1, we may consider theHermitian code obtained by evaluating the functions M ( s ) = { X i Y j | iq + j ( q + 1) ≤ s, ≤ i ≤ q − , ≤ j ≤ q − } . In order to find a subcode of the Hermitian code with a given locality, proceedas follows: Let g ( x ) be a polynomial of degree r + 1. Let A , A , . . . , A q − r be apartition of F q into multiplicative cosets of F ∗ q where g ( X ) is constant on each A i . Likewise let g ( Y ) be a polynomial of degree r + 1. Let B , B , . . . , B qr be a partitiion of { γ | γ q + γ = 0 } where g ( Y ) is constant on each B j . For fixed s , s , the code obtained by evaluating L s ,s ( s ) = X i g ( X ) i Y j g ( Y ) j | ≤ i ≤ r , i (cid:54) = s , ≤ j ≤ r , ≤ i + ( r + 1) i ≤ q − , ≤ j + ( r + 1) j ≤ q − is a subcode of the Hermitian code of degree s which also has locality r and lo-cality r with availability 2. In this case we obtain the codes over F with locality , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , F , which have parameters: [64 , , , , , , , , , , , , F q is the set of F q –rational points, we evaluate the intersection of aproduct code of two LRC codes with the Suzuki code of length q . The vanishingideal of the full affine plane is the ideal spanned by X q − X, Y q − Y . In this case,the Suzuki code is obtained by taking the polynomials of low order at infinity andevaluating at the q rational points. One can also take the remainders modulo X q + X, Y q + Y to determine the dimension of the code instead. Hence the Suzukicode is obtained by evaluating L ( s ) = { X a Y b U c V d mod X q + X, Y q + Y | aq + b ( q + q )+ c ( q +2 q )+ d ( q +2 q +1) ≤ s } . In this case, the optimal LRC codes of locality 3 and length 8 have param-eters: [8 , , , , , , , , , , , , , ,
21] code.From the Suzuki code construction, after imposing additional LRC conditions,we get a [64 , ,
36] code with the same locality and availability parameters.In the following table, we compare some Suzuki LRC codes with some RSproduct LRC codes. Both have the same length, symbols, locality and availabil-ity. Note that we were able to improve on most of the Product code constructions,except for [64 , , q = 27, availability is3 and r = r = r = 8. Please recall that the Ree curve R q may be describedby the equation R q : (cid:40) y q − y = x q ( x q − x ) z q − z = x q ( x q − x )where q = 3 s , q = 3 q , and s ∈ N . The valuation of x at infinity is q , thevaluation of y at infinity is q + q and the valuation of z at infinity is q +2 q . If G represents the pole at infinity of the Ree curve, and D is the divisorcorresponding to the F –affine points of the Ree curve, then the code C ( sG, D )is the algebraic geometry code obtained by evaluating all functions having polesonly at infinity of order ≤ s . We shall compare LRC subcodes of C ( sG, D ) withproduct codes of Reed–Solomon codes. We shall use a particular Tamo–Bargconstruction [21] for F . In this case, our sets will be places where the trace is odes with locality from cyclic extensions of Deligne-Lusztig curves 15[ s , s ] Suzuki code Suzuki code with locality and availability[1 ,
0] [64 , ,
64] [64 , , ,
0] [64 , ,
56] [64 , , ,
0] [64 , ,
54] [64 , , ,
0] [64 , ,
51] [64 , , ,
0] [64 , ,
48] [64 , , ,
0] [64 , ,
46] [64 , , ,
0] [64 , ,
44] [64 , , ,
0] [64 , ,
40] [64 , , ,
0] [64 , ,
38] [64 , , ,
0] [64 , ,
36] [64 , , ,
0] [64 , ,
32] [64 , , ,
0] [64 , ,
31] [64 , , ,
0] [64 , ,
30] [64 , , ,
0] [64 , ,
28] [64 , , ,
0] [64 , ,
27] [64 , , ,
0] [64 , ,
26] [64 , , ,
0] [64 , ,
24] [64 , , ,
0] [64 , ,
22] [64 , , ,
0] [64 , ,
20] [64 , , ,
0] [64 , ,
19] [64 , , ,
0] [64 , ,
16] [64 , , ,
0] [64 , ,
14] [64 , , ,
0] [64 , ,
12] [64 , , ,
0] [64 , ,
11] [64 , , ,
0] [64 , ,
8] [64 , , ,
0] [64 , ,
7] [64 , , ,
0] [64 , ,
6] [64 , , ,
0] [64 , ,
4] [64 , , ,
0] [64 , ,
2] [64 , , Table 1.
Comparison of parameters of codes from the Suzuki curve using a standardAG code construction and those with locality and availability6 G. L. Matthews and F. Pi˜neroSuzuki code with locality and availability Comparable Product code[64 , ,
64] [64 , , , ,
56] [64 , , , ,
54] [64 , , , ,
51] [64 , , , ,
48] [64 , , , ,
46] [64 , , , ,
44] [64 , , , ,
40] [64 , , , ,
38] [64 , , , ,
36] [64 , , , ,
32] [64 , , , ,
31] [64 , , , ,
30] [64 , , , ,
28] [64 , , , ,
27] [64 , , , ,
26] [64 , , , ,
24] [64 , , , ,
22] [64 , , , ,
20] [64 , , , ,
19] [64 , , , ,
16] [64 , , , ,
14] [64 , , , ,
12] [64 , , , ,
11] [64 , , , ,
8] [64 , , , ,
7] [64 , , , ,
6] [64 , , , ,
4] [64 , , , ,
2] [64 , , Table 2.
Comparison of parameters of codes from the Suzuki curve with locality andavailability and product codesodes with locality from cyclic extensions of Deligne-Lusztig curves 17 constant A = { a ∈ F | a + a + a = 0 } , A = { a ∈ F | a + a + a = 1 } and A = { a ∈ F | a + a + a = 2 } . The polynomial L ( T ) = T + T + T isconstant on the recovery sets A , A , A and the evaluation of the polynomials in { L ( T ) i T j | ≤ i ≤ , ≤ j ≤ , i + j ≤ k } gives a subcode of the Reed–Solomoncode RS ( F , k ) which is an LRC with locality 8.The possible Reed–Solomon codes with locality 8 and length 27 of this formare: [27 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , L = { X i Y j Z l | ≤ i, j, l ≤ } . In the F .The subcode of C ( sG, D ) is obtained by evaluating the monomials in L ( s ) = { X a Y b Z c ∈ L | aq + b ( q + q ) + c ( q + 2 q ) ≤ s, ≤ a, b, c ≤ q − } . The subcode of C ( sG, D ) with locality 8 and availability 3 is given by evaluatingpolynomials of the form L ( X ) a X a L ( Y ) b Y b L ( Z ) c Z c where 0 ≤ a , b , c ≤ , ≤ a , b , c ≤ L ( s ). Note that depending on the parameters of the codes wemight find better Reed–Solomon product codes as LRCs or better LRC subcodesfrom the Ree curve.For example, comparing codes with minimum distance 600 we get an LRCwith parameters [19683 , , , , C ( sG, D ). However, comparing codeswith dimension 200 we get a [19683 , , , , C ( sG, D ). In this casenote that for the same replication sets A , A and A , the dual code of an LRCwith locality 8 is generated by evaluating { L ( T ) i | ≤ i ≤ } . If we extend thisto F we can get a code with locality 8 and availability 3 as the dual code ofthe evaluation of { L ( X ) i L ( Y ) j L ( Z ) k | ≤ i ≤ } . In order to find subcodes of C ( sG, D ) with locality 8 and availability 3 we consider how many of the functions { L ( X ) i L ( Y ) j L ( Z ) k | ≤ i ≤ } also have weight s . The dimension of the LRCis found by computing the dimension of L ( s ) + { L ( X ) i L ( Y ) j L ( Z ) k | ≤ i ≤ } .In this case we have found a [19683 , , , , C ( sG, D ) ⊥ .We have found instances in which the Reed–Solomon product codes are betterthan the LRC from the AG codes. Likewise, we have found cases in which theAG LRCs outperform the Reed–Solomon codes. Further improvements could bepossible as knowledge of the Riemann-Roch spaces of the Ree curve improves. References
1. Barg, A., Haymaker, K., Howe, E., Matthews, G. L., Varilly-Alvarado A.: Locallyrecoverable codes from algebraic curves and surfaces, in Algebraic Geometry forCoding Theory and Cryptography, E.W. Howe, K.E. Lauter, and J.L. Walker,Editors, Springer, 2017, pp. 95–126. doi: 10.1007/978-3-319-63931-4 42. Ballentine, S., Barg, A., Vl˘adut¸, S.: Codes with hierarchical locality from coveringmaps of curves, IEEE Transactions on Information Theory, vol. 65, no. 10, pp.6056–6071, Oct. 2019. doi: 10.1109/TIT.2019.29198303. Eid, A., Duursma, I.: Smooth embeddings for the Suzuki and Ree curves. Al-gorithmic arithmetic, geometry, and coding theory, vol. 637, 251-291, 2015. doi:10.1090/conm/637/127634. Eid, A., Hasson, H., Ksir, A. , Peachey, J.: Suzuki-invariant codes from theSuzuki curve. Designs, Codes and Cryptography, vol. 81, pp. 413– 425, 2016. doi:10.1007/s10623-015-0164-55. Giulietti, M., Korchmros, G.: A new family of maximal curves over a finite field,Mathematische Annalen, vol. 343, article 229, 2009. doi: 10.1007/s00208-008-0270-z6. Giulietti, M., Montanucci, M., Quoos, L., Zini, G.: On some Galois covers of theSuzuki and Ree curves, Journal of Number Theory, vol. 189, pp. 220–254. doi:10.1016/j.jnt.2017.12.005.7. Guruswami, V., Jin, L., Xing, C.: Constructions of maximally recoverable localreconstruction codes via function fields, International Colloquium on Automata,Languages, and Programming, 2019. doi: 10.4230/LIPIcs.ICALP.2019.688. Hansen, J.P., Stichtenoth, H.: Group codes on certain algebraic curves with manyrational points, Applicable Algebra in Engineering, Communication and Comput-ing, vol. 1, pp. 67–77, 1990. doi: 10.1007/BF018108499. Haymaker, K., Malmskog, B., Matthews, G. L.: Locally recoverable codes withavailability t ≥≥