Bounds on List Decoding of Linearized Reed-Solomon Codes
BBounds on List Decoding ofLinearized Reed–Solomon Codes
Sven Puchinger, Johan Rosenkilde
Department of Applied Mathematics and Computer Science,Technical University of Denmark (DTU), Lyngby, DenmarkEmail: [email protected], [email protected]
Abstract —Linearized Reed–Solomon (LRS) codes are sum-rank metric codes that fulfill the Singleton bound with equality. Inthe two extreme cases of the sum-rank metric, they coincide withReed–Solomon codes (Hamming metric) and Gabidulin codes(rank metric). List decoding in these extreme cases is well-studied,and the two code classes behave very differently in terms of listsize, but nothing is known for the general case. In this paper, wederive a lower bound on the list size for LRS codes, which is, for alarge class of LRS codes, exponential directly above the Johnsonradius. Furthermore, we show that some families of linearizedReed–Solomon codes with constant numbers of blocks cannot belist decoded beyond the unique decoding radius.
Index Terms —Sum-Rank Metric, Linearized Reed–SolomonCodes, List Decoding
I . I
N T R O D U C T I O N
The sum-rank metrics is a fairly recent family of metrics:two vectors of length n are split into (cid:96) blocks each and theirsum-rank distance is the sum of the rank distances of the blockpairs. For (cid:96) = n , this is simply the Hamming metric, while for (cid:96) = 1 it is the rank metric. This family was introduced in 2010[1], motivated by applications in multi-shot network coding.Other applications are distributed storage [2], other aspects ofnetwork coding [3], and space-time codes [4]. There are severalcode constructions and decoding algorithms for codes in thesum-rank metric [3], [5]–[15].The class of linearized Reed–Solomon (LRS) codes wereintroduced in [10], and for a given sum-rank metric, there is alarge family of LRS codes meeting the Singleton bound for thatmetric. When the sum-rank metric is actually the Hammingmetric, the corresponding family of LRS codes is the well-known Reed–Solomon (RS) codes [16], and when it is the rankmetric, the corresponding LRS codes are the Gabidulin codes[17]–[19]. Also the restriction on the number of blocks (cid:96) andthe block size n(cid:96) of an LRS code are a mix of the two extremecases: If the code is defined over F q m , where the rank is takenw.r.t. the subfield q , then LRS codes require (cid:96) < q and n(cid:96) ≤ m .The topic of this paper is the list size [20], [21] of LRScodes, which is the maximum number of codewords in a ballof given radius in the sum-rank metric (maximized over allpossible centers of the ball). For the extreme cases of LRScodes, the combinatorial list decoding problem is well-studied:RS codes in the Hamming metric ( (cid:96) = n ) have a polynomiallist size up to the relative decoding radius J R = 1 − √ R , where R the rate, known as the Johnson radius [22]. Understandingthe list size beyond J R is a long-standing open problem. Weknow that “most” RS codes allow list decoding beyond J R [23].On the other hand, [24] and [25] showed that the list size is This work has been supported by the European Union’s Horizon 2020research and innovation programme under the Marie Sklodowska-Curie grantagreement no. 713683. exponential below the channel capacity for RS codes whosedimension grows exponentially slower than the length.For Gabidulin codes in the rank metric ( (cid:96) = 1 ), Wachter-Zeh[26] and Raviv–Wachter-Zeh [27] adapted the arguments of[24], [25] to show that all
Gabidulin codes with n = m hasan exponential list-size starting from J R , and that some haveexponential list size already from half the minimum distance.For the latter result, they gave explicit constructions with fixedrates ≥ . . A. Contributions
We basically extend the known list size results for Gabidulincodes in the rank metric to almost all LRS codes in the sum-rankmetric.More specifically, in Section III, we show that all
LRS codeshave exponential list size in n above a specific radius. For LRScodes with (cid:96) ∈ o ( n ) and of smallest possible field extensiondegree m = n(cid:96) , this radius is the Johnson radius J R . For certainfamilies of LRS codes with (cid:96) ∈ Θ( n ) , we also obtain a fixedrelative decoding radius slightly beyond J R . For RS codes (cid:96) = n , our bound is the same as in [25].In Section IV, we show that some families of LRS codeshave exponential list size directly above half the minimumdistance. For (cid:96) = 1 , , . . . , the constructed families may haverates ≥ . , . , . , . , . , . , . . . , and asymptotically ≥ − Θ( / √ (cid:96) ) . This result extends [27] to LRS codes with aconstant number of blocks (cid:96) .The results indicate that LRS codes in the studied parameterranges behave similarly to Gabidulin codes ( (cid:96) = 1 ) in terms oflist decodability. Fig. 1 illustrates some of the results for LRScodes with m = n(cid:96) . . . . . . . . . Unique decodingExponential list size for (cid:96) ∈ o ( n ) (any code)Exponential list size for R ≥ − Θ( / √ (cid:96) ) (some codes) Rate R = kn R e l a ti v e D ec od i ng R a d i u s τ n Figure 1. Illustration of the results for LRS codes with block size n(cid:96) = m ,where m is the extension degree of the underlying field extension. a r X i v : . [ c s . I T ] F e b I . P
R E L I M I N A R I E S
Let q be a prime power and m be a positive integer. Wedenote by F q the finite field of size q and by F q m its extensionfield of extension degree m . We will make use of the fact that F q m is an m -dimensional F q -vector space, which means thatthe rank rk F q ( x ) := dim F q (cid:104) x , . . . , x η (cid:105) F q of a vector x ∈ F ηq m is well-defined. A. Sum-Rank Metric
Throughout this paper, let (cid:96), η, n ∈ Z > such that n = (cid:96)η .We say that n is the code length , (cid:96) is the number of blocks ,and η is the block size . Such a triple together with a finite field F q m induces a sum-rank metric defined as follows. Definition 1.
The ( (cid:96) -)sum-rank weight on F nq m is defined as wt SR ,(cid:96),q : F nq m → Z ≥ , x (cid:55)→ (cid:80) (cid:96)i =1 rk F q ( x i ) , where we write x = (cid:2) x | x | . . . | x (cid:96) (cid:3) with x i ∈ F ηq m . Further-more, the ( (cid:96) -)sum-rank distance is defined as d SR ,(cid:96),q : F nq m × F nq m → Z ≥ , [ x , x ] (cid:55)→ wt SR ,(cid:96),q ( x − x ) . For (cid:96) = 1 , the sum-rank metric coincides with the rank metricand for (cid:96) = n , it is the Hamming metric.The ball with center r ∈ F nq m and radius τ w.r.t. the (cid:96) -sum-rank metric is defined as B τ,(cid:96) ( r ) := (cid:8) x ∈ F nq m : d SR ,(cid:96),q ( x , r ) ≤ τ (cid:9) . If C ⊆ F nq m is a code (i.e. any subset of F nq m ), and τ is anypositive integer at most n , then the τ - list size of C (wrt. the (cid:96) -sum-rank) is defined as: L ( C , τ ) := max r ∈ F qm |C ∩ B τ,(cid:96) ( r ) | . B. Conjugacy in a Finite Field
Let ψ ∈ Gal( F q m / F q ) be an element of the Galois group ofthe field extension F q m / F q . This means that ψ = φ sq is a powerof the Frobenius automorphism φ q : F q m → F q m , a (cid:55)→ a q .The fixed field F ψq m of ψ is the set { a ∈ F q m : ψ ( a ) = a } .Note that F ψq m is indeed a field and F q ⊆ F ψq m ⊆ F q m . Further F ψq m = F q exactly when ψ generates the group Gal( F q m / F q ) ,which is exactly when ψ = φ sq for an s satisfying gcd( m, s ) = 1 .The norm map w.r.t. ψ is the multiplicative homomorphism N ψ : F ∗ q m → ( F ψq m ) ∗ ,a (cid:55)→ [ F qm : F ψqm ] − (cid:89) i =0 ψ i ( a ) . We define conjugacy as follows:
Definition 2 ( [28], [29]) . Two elements a, b ∈ F ∗ q m are conjugates w.r.t. ψ if they have the same norm N ψ ( a ) = N ψ ( b ) . This defines an equivalence relation on F ∗ q m with | F ψq m | − conjugacy classes of size q m − | F ψqm |− each. By Hilbert’s Theorem90, then a and b are conjugates if and only if there is a c ∈ F q m such that a = bψ ( c ) c − .The norm, and hence also conjugacy, turns out not todepend on ψ but only on F ψq m . Throughout this paper, σ ∈ Gal( F q m / F q ) will be chosen to have fixed field F q , in whichcase N σ ( a ) = N φ q ( a ) = a q + ... + q m − , and conjugacy wrt. σ is the commonly used notion of conjugacy for the extension F q m / F q . C. Skew Polynomials
Skew polynomials were first introduced by Ore in [30] in aquite general setting. In this paper, we use the following specialcase (in particular, we do not use derivations):
Definition 3.
The ring of skew polynomials F q m [ x ; σ ] is a setof formal polynomials (cid:40) f = d (cid:88) i =0 f i x i : f i ∈ F q m , d ∈ Z ≥ (cid:41) equipped with ordinary (component-wise) addition f + g = (cid:88) i ≥ ( f i + g i ) x i and the multiplication rule x · a = σ ( a ) · x, extended to polynomials by associativity and distributivity. Multiplication in F q m [ x ; σ ] is non-commutative whenever σ (cid:54) = id . The degree of a skew polynomial is defined as deg f := (cid:40) max { i : f i (cid:54) = 0 } , if f (cid:54) = 0 , −∞ , otherwise.For an integer n , we denote by F q m [ x ; σ ] The codes studied in this paper are evaluation codes of skewpolynomials, where the used evaluation map is the generalizedoperator evaluation [31], defined as follows. Let f ∈ F q m [ x ; σ ] and a ∈ F ∗ q m . Define f ( · ) a : F q m → F q m ,β (cid:55)→ (cid:88) i ≥ f i σ i ( β ) N i ( a ) , where N i ( a ) := (cid:81) i − j =0 σ j ( a ) . Due to the F q -linearity of σ , themap f ( · ) a is F q -linear for a fixed a .For a := [ a , . . . , a (cid:96) ] ∈ F (cid:96)q m and β := [ β , . . . , β η ] ∈ F ηq m ,define the multi-point evaluation map ev a , β ( · ) : F q m [ x ; σ ] → F nq m f (cid:55)→ (cid:104) f ( β ) a , . . . , f ( β η ) a , f ( β ) a , . . . , f ( β η ) a (cid:96) (cid:105) . We will use such an evaluation map only when a and β satisfy certain criteria, which we give a name: Definition 4. A pair ( a , β ) ∈ F (cid:96)q m × F ηq m is said to be an evaluation pair (wrt. σ ) if the elements of a are in distinct con-jugacy classes, and the elements of β are linearly independentover F q . The following is well-known. Lemma 1 (Collection of results in [10], or [12, Proposi-tion 1.3.7]) . Let ( a , β ) ∈ F (cid:96)q m × F ηq m be an evaluation pair.Then, • the restricted mapping ev a , β ( · ) | F qm [ x ; σ ] The number of matrices in F m × ηq of rank t ≤ max { η, m } isgiven by [32] NM q ( m, η, t ) = t − (cid:89) i =0 (cid:0) q m − q i (cid:1) (cid:0) q η − q i (cid:1) q t − q i , and we can bound it from below and above by [32], [33] γ − q q t ( m + η − t ) ≤ NM q ( m, η, t ) ≤ γ q q t ( m + η − t ) , where γ q := (cid:81) ∞ i =1 (1 − q − i ) − is a constant depending onlyon q , which is monotonically decreasing in q with a limit of ,and e.g. γ ≈ . , γ ≈ . , and γ ≈ . .The number of vectors in F nq m of (cid:96) -sum-rank weight t isdefined as N q,η,m ( t, (cid:96) ) , and we have N q,η,m ( t, (cid:96) ) = (cid:88) t ∈T t,(cid:96),µ (cid:96) (cid:89) i =1 NM q ( m, η, t i ) . We make use of the following lower bound on N q,η,m ( t, (cid:96) ) . Lemma 2 ([33]) . We have N q,η,m ( t, (cid:96) ) ≥ (cid:40) q t ( η + m − t(cid:96) ) γ − (cid:96)q , (cid:96) | t,q t ( η + m − t(cid:96) ) − (cid:96) γ − (cid:96)q , (cid:96) (cid:45) t. I I I . L R S C O D E S W I T H E X P O N E N T I A L L I S T S I Z E A B O V E T H E ( H A M M I N G ) J O H N S O N R A D I U S Our list-size lower bounds are based on the following simplecounting strategy, originally used in [24] for the Hammingmetric and expounded in [25]: Lemma 3. Consider (cid:96), η, n which induces a sum rank metricon F nq m such that η ≤ m and (cid:96) < q , and σ an automorphism of F q m with fixed field F q , and C = C ( a , β ) LRS [ n, k ] a linearized RScode. Let τ be a positive integer less than d = n − k +1 . Let S ⊆ F q m [ x ; σ ] Divide S into disjoint subsets such that all polynomialsin each subset have the same coefficients of degree k, k + 1 , . . . , n − . Note that there are at most q ms such subsets.Let S (cid:48) ⊂ S be such a subset of maximal cardinality. By thePigeonhole principle, then L := | S (cid:48) | ≥ | S | q − ms . Write S (cid:48) = { f , . . . , f L } , and set r := ev a , β ( f ) , c i := ev a , β ( f i − f ) ∀ i = 1 , . . . , L. Then, d SR ,(cid:96),q ( r , c i ) = wt SR ,(cid:96),q (ev a , β ( f i )) ≤ τ by definition of S and deg( f i − f ) < k since the n − k top-mostcoefficients of the f i are the same. Hence, c , . . . , c L ∈ C ∩ B τ,(cid:96) ( r ) , which proves the claim.The game is now to construct sets S such that the trade-offbetween | S | and s makes the value | S | q − ms as large as possible.Our first bound is achieved by choosing S to be all possiblesatisfactory polynomials: Theorem 4. Let (cid:96), m, n, η, k, d be valid parameters of alinearized RS code C and τ < d . Then L ( C , τ ) ≥ q m + τ ( m + η ) − τ (cid:96) − md U − , where U = γ (cid:96)q if (cid:96) | τ and U = ( q / γ q ) (cid:96) otherwise.Proof. Consider the set S of all linearized polynomials ofdegree less than n such that their ev a , β -image has (cid:96) -sum-rankweight at most τ . Since ev a , β : F q m [ x ; σ ] Fix a prime power q and some real number ε > ,and an infinite subset N ⊂ N . Let {C n } n ∈ N be a family oflinearized RS codes such that C n has length n over an extensionfield of F q . For each such n ∈ N , choose a decoding radius τ greater or equal to (cid:96)m + n − (cid:113)(cid:0) (cid:96)m + n (cid:1) − (cid:96) (cid:0) + log q ( γ q ) (cid:1) − (cid:96)m ( d − − εn ) , where the code parameters correspond to the respective code C n . Then L ( C n , τ n ) ∈ Ω( q εn/(cid:96) ) . Corollary 6. Fix a rate R . In the context of Corollary 5, assumethat C n has k = (cid:98) Rn (cid:99) and m = η = n/(cid:96) , and where (cid:96) ∈ o ( n ) holds for the family. Then there is a sequence of decoding radii τ n such that τ n /n → − √ R + ε and L ( C n , τ n ) ∈ ω ( q εn ) .roof. Since (cid:96) ∈ o ( n ) , we may choose as τ n the least integergreater than n (1 − √ R + ε ) which is divisible by (cid:96) . Then wemay set U = γ (cid:96)g in Theorem 4, and observe log q ( L ( C n , τ n )) : log q ( L ( C n , τ n )) ≥ (cid:0) (cid:96) ( n + 2 τ n − τ − nd ) − (cid:96) log q ( γ q )= n (cid:96) (2(1 − √ R + ε ) − (1 − √ R + ε ) − (1 − √ R )) − (cid:96) log q ( γ q )= εn (cid:96) (2 √ R − ε ) − (cid:96) log q ( γ q )= εn (cid:16) n(cid:96) (2 √ R − ε ) − (cid:96)εn log q ( γ q ) (cid:17) Since (cid:96) ∈ o ( n ) then n(cid:96) → ∞ and (cid:96)εn → , and hence L ( C n , τ n ) ∈ ω ( q εn ) .Recall that m = η is the minimal possible field extensionfor a linearized RS code. Then Corollary 6 implies that as longas (cid:96) does not grow as fast as n , the list size grows exponentialabove the Johnson radius.The following corollary shows that slightly beyond theJohnson bound, we may even conclude an exponential list sizebound for most families of LRS codes where (cid:96) grows linearlyin n (i.e. very Hamming-like codes): Corollary 7. Fix a rate R and a constant a ∈ ]0; 1[ such that ζ := a ( / + log q ( γ q )) < R . In the context of Corollary 5,assume that C n has k = (cid:98) Rn (cid:99) and m = η = n/(cid:96) , and where (cid:96) < an , and choose ε > √ R − √ R − ζ ∈ R + . Then there is asequence of decoding radii τ n such that τ n /n → − √ R + ε and L ( C n , τ n ) ∈ ω ( b n ) , for some real number b > .Proof. We set U = ( q / γ g ) (cid:96) = q (cid:96)ζ/a in Theorem 4 andobserve log q ( L ( C n , τ n )) :log q ( L ( C n , τ n )) ≥ (cid:96) ( n + 2 τ n − τ − nd ) − (cid:96)ζa ≥ εna (2 √ R − ε ) − nζa = n δa , where δ := − ε + 2 √ Rε − ζ . Then δ > is assured by thechoice of ε . Choosing b = q δ/a completes the proof.I V. S O M E L R S C O D E S C A N N O T B E L I S T D E C O D E DAT A N Y R A D I U S In the following, if f ∈ F q m [ x ; σ ] , then supp( f ) denotes theset of exponents for which f has a non-zero monomial. E.g. if f = x + 3 , then supp( f ) = { , } . A. Sparse Skew Polynomials Definition 6. Consider (cid:96), η, n which induces a sum rank metricon F nq m such that η ≤ m and (cid:96) < q , and σ an automorphismof F q m with fixed field F q , and let ( a , β ) ∈ F (cid:96)q m × F ηq m be anevaluation pair. Let ≤ τ ≤ n be a decoding radius and letfurther g ∈ Z > be a sparsity index. We define the set R ( a , β ) g ( σ, τ ) := (cid:110) f ∈ F q m [ x ; σ ] Let (cid:96), η, n, m, τ, a , β be chosen as in Definition 6with the additional restrictions (cid:96) | τ , g | m , η = m , g | τ , and β of the form β := [ α γ , . . . , α γ g , α γ , . . . , α η/g γ g ] ∈ F ηq m , Then, the β i are linearly independent over F q and we have (cid:12)(cid:12)(cid:12) R ( a , β ) g ( σ, τ ) (cid:12)(cid:12)(cid:12) ≥ q τg ( m + η − τ(cid:96) ) γ − (cid:96)q g . Proof. Let f ∈ F q m [ y ; σ g ] have degree < n/g and define ˜ f ∈ F q m [ x ; σ ] Assume the same setting as in Theorem 8 andchoose k > n − τ . Let C be the linearized RS code ofparameters [ n, k, d ] q m , block size η , number of blocks (cid:96) , andevaluation point vectors a and β . Then L ( C , τ ) ≥ q m + τg ( η − m − τ(cid:96) ) γ − (cid:96)q g . (2) Proof. Combine Theorem 8 with Lemma 3, and note that thepolynomials in R ( a , β ) g ( q, τ ) may only have up to τg non-zeromonomials of degree k = n − τ or higher. B. Families of LRS Codes with Exponential List Size Above theUnique Decoding Radius The following construction gives families of LRS codes withexponential list size directly above the unique decoding radius.For (cid:96) = 1 (rank metric), the construction coincides with theexample families constructed in [27, Section IV]. Construction 10. Fix (cid:96), C, D ∈ Z > with (cid:96) < q , and (cid:96) | C , and C > max { D , D } . Define a family of LRS codes {C g } g ∈ Z > , (cid:96) | g by choosing C g to have the code parameters • n = Cg • k = n − Dg + 1 • m = η = Cg(cid:96) and a ∈ ( F ∗ q m ) (cid:96) and β ∈ F ηq m chosen in any way to satisfyTheorem 8. The following theorem shows that the code families inConstruction 10 an exponential list size directly above halfthe minimum distance (cid:98) n − k (cid:99) . Theorem 11. Let (cid:96), C, D, {C g } g ∈ Z > , (cid:96) | g be chosen as inConstruction 10. Then,1) For g → ∞ , the rate of C g converges to R ( C g ) → − DC > − √ C . 2) For g → ∞ , we have L ( C g , (cid:98) n − k (cid:99) + 1) ∈ Ω ( q cn ) , where c is a positive constant that depends only on (cid:96), C, D .Proof. Ad 1): We have R ( C g ) = kn = 1 − DC − Cg → − DC . The inequality follows by C ≥ D + 1 . Ad 2): First note that k > n − τ with τ g := (cid:98) n − k (cid:99) + 1 = Dg . Since by the choice of n, m, η, k, τ, α , a , all conditions ofCorollary 9 are fulfilled, then L ( C g , τ g ) ≥ q m + τg ( η − m − τ(cid:96) ) γ − (cid:96)q g = q g(cid:96) ( C − D ) γ − (cid:96)q g . The claim follows due to γ − (cid:96)q g → , (cid:96) being a constant, and C − D > .We can choose a family as in Construction 10 with a resultingrate R arbitrarily close to by setting D = 1 and C anypositive multiple of (cid:96) greater than . 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