Bounds and Genericity of Sum-Rank-Metric Codes
aa r X i v : . [ c s . I T ] F e b Bounds and Genericity of Sum-Rank-Metric Codes
Cornelia Ott , Sven Puchinger , Martin Bossert Institute of Communications Engineering, Ulm University, Germany Department of Applied Mathematics and Computer Science, Technical University of Denmark (DTU), Denmark
E-mail: [email protected], [email protected], [email protected]
Abstract —We derive simplified sphere-packing and Gilbert–Varshamov bounds for codes in the sum-rank metric, which canbe computed more efficently than previous ones. They give rise toasymptotic bounds that cover the asymptotic setting that has notyet been considered in the literature: families of sum-rank-metriccodes whose block size grows in the code length. We also providetwo genericity results: we show that random linear codes achievealmost the sum-rank-metric Gilbert–Varshamov bound with highprobability. Furthermore, we derive bounds on the probabilitythat a random linear code attains the sum-rank-metric Singletonbound, showing that for large enough extension field, almost alllinear codes achieve it.
Index Terms —sum-rank metric, Gilbert–Varshamov bound,sphere-packing bound
I. I
NTRODUCTION
The sum-rank metric is a mix of the Hamming and rankmetric. It was first introduced in 2010 [1], motivated by multi-shot network coding. Since then, many code constructions anddecoding algorithms for sum-rank-metric codes have been pro-posed [2]–[13]. Some of these codes have found applicationsin distributed storage [14], further aspects of network coding[9], and space-time codes [15].In two extreme cases, the metric coincides with the Ham-ming and the rank metric, respectively, and thus many fun-damental bounds on the code parameters are known [16]–[19] for these two cases. Although the sum-rank metric hasbeen studied since 2010, only very recently, Byrne, Gluesing-Luerssen, and Ravagnani [13] presented (among many otherfundamental results) a sphere-packing and Gilbert–Varshamovbound for sum-rank metric codes. They also presented asymp-totic versions of the bounds on sum-rank metric codes forbounded block sizes and growing number of blocks. Thebounds for finite parameters depend on the sum-rank-metricball size, which is super-polynomial to compute using thepresented formula.Furthermore, it is well-known that random codes in theHamming and rank metric [19] achieve the respective Gilbert–Varshamov bound with high probability, hence codes attainingthese bounds are the generic case. Bounds on the probabilitythat random codes fulfill the Hamming or rank-metric Single-ton bound with equality (called maximum distance separable(MDS) or maximum rank distance (MRD) codes, respectively),have been derived in [20] and [21], respectively. No such resultis known for the sum-rank metric, where codes attaining theSingleton bound [7] are called maximum sum-rank distance(MSRD) codes.In this paper, we extend these the results by Byrne et al.,as well as the genericity results from the Hamming and rankmetric, as follows. We present variants of the sphere-packing(SP) and Gilbert–Varshamov (GV) bound for linear codes anddraw the connection to a recent algorithm to compute sum-rank-metric sphere sizes [22], which allows to compute thebounds in polynomial time. Using lower and upper bounds on the sum-rank-metric ball size, we derive simplified boundsthat can be computed even more efficiently. These simplifiedbounds also induce asymptotic variants of the two bounds,which extend the results in Byrne et al. by covering also thecase of growing block size.Furthermore, we present the following genericity results: weshow that random linear codes achieve almost the sum-rankGV bound with high probability and derive two bounds on theprobability that a random code is MSRD. The bounds smoothlyinterpolate between the known bounds in the Hamming andrank metric and show that MSRD codes are generic forgrowing extension degree of the underlying field.II. P
RELIMINARIES
We use a similar notation as in [22]. Let q be a prime powerand m, n, ℓ, η positive integers. We denote by F q a finite fieldwith q elements and by F q m its extension field. The codes weconsider in this paper are subsets of F nq m , where each vector x = [ x | . . . | x ℓ ] ∈ F nq m consists of ℓ blocks x , . . . , x ℓ ∈ F ηq m of length η . Therefore we assume n = ℓ · η . Since F q m is a avectorspace over F q of dimension m , a vector x i ∈ F ηq m canalso be represented as a matrix X i ∈ F m × ηq , hence the rankweight of x i is defined as wt Rk ( x i ) := dim F q h x , . . . , x η i ,which is equal to the rank of the matrix X i . Clearly itholds for x i ∈ F ηq m that wt Rk ( x i ) ∈ { , . . . , µ } , where µ := min { m, η } . We define sum-rank weight and the sum-rank distance of a vector x ∈ F nq m as follows. Definition 1.
Let x = [ x | . . . | x ℓ ] ∈ F nq m . The ( ℓ -)sum rankweight of x is defined as wt SR,ℓ : F nq m → N , x P ℓi =0 wt Rk ( x i ) . For two vectors x , x ′ ∈ F nq m the ( ℓ -)sum rank distance isdefined as d SR,ℓ : F nq m × F nq m → N , ( x , x ′ ) d SR,ℓ ( x , x ′ ) := wt SR,ℓ ( x − x ′ ) . The vector [wt Rk ( x ) , . . . , wt Rk ( x ℓ )] is called the weightdecomposition of x . The ( ℓ -)sum-rank distance d SR,ℓ is a metric over F nq m , thesocalled sum-rank metric . In the following we define spheresand balls in the sum-rank metric analogues to [19] and givedefintions for their volume. Definition 2.
Let τ ∈ Z ≥ with ≤ τ ≤ ℓ · µ and x ∈ F nq m .The sum-rank-metric sphere with radius τ and center x isdefined as S ℓ ( x , τ ) := { y ∈ F nq m | d SR,ℓ ( x , y ) = τ } . Analogously, we define the ball of sum-rank radius τ withcenter x by B ℓ ( x , τ ) := S τi =0 S ℓ ( x , i ) . e also define the following cardinalities: Vol S ℓ ( τ ) := |{ y ∈ F nq m | wt SR,ℓ ( y ) = τ }| , Vol B ℓ ( τ ) := P τi =0 Vol S ℓ ( i ) . Since the sum-rank metric is invariant under translation ofvectors, the volume of a sphere or ball is independent ofits center. Hence,
Vol S ℓ ( τ ) and Vol B ℓ ( τ ) are the volumesof any sphere or ball of radius τ . Unlike the extreme cases,Hamming and rank metric, it is quite involved to compute thesevolumes in general. The formula given in [13] consists of a sumwhose number of summands may grow super-polynomiallyin τ , depending on the relative size of ℓ and η . In [22], adynamic-programming algorithm was given, which computesthe volumes in polynomial time.We define a linear sum-rank metric code as follows. Definition 3.
A linear sum-rank metric code C over F q m oflength n and dimension k is an F q m -vector space C ⊂ F nq m with dim F qm ( C ) = k . Hence, the cardinality of the code is |C| = q mk . Each codeword c = [ c | . . . | c ℓ ] ∈ C consists of ℓ blocks c i ∈ F ηq m of length η . The minimum ( ℓ )-sum-rankdistance d is defined as d := min c = c ′ ∈C { d SR,ℓ ( c , c ′ ) } = min c ∈C { wt SR,ℓ ( c ) } . We denote such a code by C ( n, k, d ) . The sum-rank weight of a vector x ∈ F nq m is at most itsHamming weight. This implies the following Singleton boundin the sum-rank metric. Theorem 1 ([7, Proposition 34]) . Let C ( n, k, d ) be a linearsum-rank metric code. Then it holds d ≤ n − k + 1 . Codes that fulfill this bound with equality are called maxi-mum sum-rank distance codes (MSRD codes). In [7, Theorem4] it is shown, that the therein defined
Linearized Reed–Solomon (LRS) codes are MSRD codes. The code parametersof LRS codes are restricted by ℓ < q and η ≤ m . It isparticularly interesting to know bounds on the code parametersfor cases in which these restrictions are not met.We define the set τ t,ℓ,µ := n t = ( t , . . . , t ℓ ) | P ℓi =1 t i = t, t i ≤ µ ∀ i o , which corresponds combinatorially to the set of ordered parti-tions with bounded number of summands and bounded sum-mands. We will extensively use the number of such partitionsthroughout the paper. By common combinatorical methods, weget | τ t,ℓ,µ | = P ηl =0 ( − l (cid:0) ℓl (cid:1)(cid:0) t + ℓ − ( η +1) lℓ − (cid:1) ≤ (cid:0) t + ℓ − ℓ − (cid:1) (1)(see also [23, Lemma 1.1]). The upper bound (cid:0) t + ℓ − ℓ − (cid:1) can alsobe easily derived by a stars- & -bars argument.III. B OUNDS IN SUM - RANK METRIC
In this section, we present bounds on sum-rank-metric codes.The first two subsections contain slight reformulations, for thecase of linear codes, of the SP and GV bounds presented in[13]. We also state the (polynomial) complexity of computingthe bounds if the efficient dynamic-programming method in[22] is used to compute
Vol S ℓ ( τ ) , instead of the formulain [13], which has super-polynomially many summands. Themain results of this section are the simplified and asymptoticSP and GV bounds in Section III-C, which we derive fromupper and lower bounds on Vol S ℓ ( τ ) . We conclude the sectionwith numerical comparisons of the bounds. A. Sphere-Packing Bound
We give an SP bound for linear codes in sum-rank metricby specializing the argument in [13] to linear codes.
Theorem 2.
For a linear sum-rank metric code C ( n, k, d ) , itholds that q mk · Vol B ℓ (cid:16)j d − k(cid:17) ≤ q mn . Furthermore, both sides of the bound can be computed incomplexity O ∼ (cid:0) ℓ d + ℓd ( m + η ) log( q ) (cid:1) using the efficientalgorithm for computing | Vol S ℓ | in [22, Theorem 5 and Algo-rithm 1].Proof: Since the minimum sum-rank distance of C is d it holds for all c , c ∈ C with c = c that B ℓ ( c , ⌊ d − ⌋ ) ∩B ℓ ( c , ⌊ d − ⌋ ) = ∅ and hence (cid:12)(cid:12)(cid:12) [ c ∈C B ℓ (cid:16) c , j d − k(cid:17)(cid:12)(cid:12)(cid:12) = P c ∈C (cid:12)(cid:12)(cid:12) B ℓ (cid:16) c , j d − k(cid:17)(cid:12)(cid:12)(cid:12) . With this fact the relation [ c ∈C B ℓ (cid:16) c , j d − k(cid:17) ⊆ F nq m leads to (cid:12)(cid:12)(cid:12) [ c ∈C B ℓ (cid:16) c , j d − k(cid:17)(cid:12)(cid:12)(cid:12) = |C| · Vol B ℓ (cid:16)j d − k(cid:17) ≤ | F nq m | . The heaviest computational step for evaluating the bound is todetermine
Vol B ℓ , which can be done by computing the spheresize Vol S ℓ ( τ ) for τ = 0 , . . . , ⌊ d − ⌋ . This can be done in theclaimed complexity by calling Algorithm 1 in [22] at most ⌊ d − ⌋ + 1 times. B. A Gilbert–Varshamov like bound
In this subsection we derive a pendant to the GV bound forthe sum-rank metric for linear codes
C ⊂ F nq m of length n anddimension k . The statement is slightly different than the GVbound in [13]: we show the existence of a linear code insteadof an arbitrary code. Theorem 3 (Gilbert–Varshamov bound) . Let F q m be a finitefield, ℓ, n, k, d ≤ µℓ be positive integers that satisfy q m ( k − · Vol B ℓ ( d − < q mn . (2) Then, there is a linear code of length n , dimension k , andminimum ℓ -sum-rank distance at least d . As in Theorem 2, wecan compute both sides of the bound in complexity O ∼ (cid:0) ℓ d + ℓd ( m + η ) log( q ) (cid:1) using the efficient algorithm for computing | Vol S ℓ | in [22, Theorem 5 and Algorithm 1].Proof: We consruct a linear code of length n , minimumsum-rank distance d and dimension k = 1 . Let c := [0 . . . ∈ F nq m and let c ∈ F nq m with wt SR,ℓ ( c ) = d . Let C := h c , c i then it holds that |C| = |{ c , α c | ∀ α ∈ F q m \ { }}| = q m .One can see, that wt SR,ℓ ( c + α c ) = d for all α ∈ F q m .Inductively, we assume that we have a linear code C ( n, k − , d ) ⊂ F nq m fulfilling (2). Since S c ∈C B ℓ ( c , d ) ( F nq m we canchoose a vector c ′ ∈ F nq m \ S c ∈C B ℓ ( c , d ) . Obviously it holdsthat d SR,ℓ ( c , c ′ ) ≥ d ∀ c ∈ C . We contruct now a vectorspace C ′ := hC ∪ c ′ i = { c + α c ′ | ∀ α ∈ F q m c ∈ C} . Forall c ∈ C , c + α c ′ ∈ C ′ with α = 0 it holds d SR,ℓ ( c , c + α c ′ ) = wt SR,ℓ ( α − ( c − c )+ c ′ ) . Since c := α − ( c − c ) ∈C one get wt SR,ℓ ( c + c ′ ) = d SR,ℓ ( c , c ′ ) ≥ d. The complexityresult follows from [22, Theorem 5]. . Simplified and Asymptotic Bounds
In this section, we derive simplified versions of the SP andGV bound based on lower and upper bounds on the volume ofa sum-rank-metric ball. These simplified bounds immediatelygive new asymptotic bounds for the two cases, in asymptoticsettings for which no asymptotic bounds are known.In order to give a lower bound on
Vol B ℓ ( t ) , we first derive inthe following lemma a lower bound for the number of m × n matrices over F q for a given rank t ≤ min { m, n } which isdenoted by NM q ( n, m, t ) . The exact number of NM q ( n, m, t ) was given in [24]: NM q ( n, m, t ) = (cid:20) nt (cid:21) q · Y t − i =0 ( q m − q i ) . We define γ q := Y ∞ i =1 (1 − q − i ) − . (3)Note that γ q is monotonically decreasing in q with a limit of , and e.g. γ ≈ . , γ ≈ . , and γ ≈ . . Lemma 1.
The cardinality of all m × n matrices over F q ofrank t ≤ min { m, n } is bounded by NM q ( n, m, t ) ≥ q ( m + n − t ) t γ − q , with γ q ≤ . defined as in (3) .Proof: The q -binomial coefficient is denoted by (cid:20) nk (cid:21) q andhere q is a prime power. In [25] the following lower boundfor the q -binomial coefficient was given: (cid:20) nt (cid:21) q ≥ q ( n − t ) t . Therefore we get NM q ( n, m, t ) ≥ q ( n − t ) t Q t − i =0 ( q m − q i ) ≥ q ( n + m − t ) t Q tj =1 (1 − q − j ) ≥ q ( n + m − t ) t γ − q . Using this bound allows us to give a lower bound on thevolume of a sphere
Vol S ℓ ( t ) = P t ∈ τ t,ℓ,µ Q ℓi =1 NM q ( η, m, t i ) containing all vectors in F nq m of sum-rank weight t . Lemma 2.
For the volumes of a sphere and of a ball withsum-rank radius t it holds: Vol B ℓ ( t ) ≥ Vol S ℓ ( t ) ≥ q ( m + η − tℓ ) t − ℓ · γ − ℓq . Proof:
We have
Vol S ℓ ( t ) = P t ∈ τ t,ℓ,µ Q ℓi =1 NM q ( η, m, t i ) ≥ max t ∈ τ t,ℓ,µ n Y ℓi =1 q ( m + η − t i ) t i γ − q o . = q ( m + η ) t · q − min t ∈ τt,ℓ,µ { P ℓi =1 t i } · γ − ℓq . We can write t as t = t ∗ · ℓ + r , with ≤ r < ℓ .Since the expression P ℓi =1 t i is minimized by the quasi-equaldecomposition: t = ( t , . . . , t ℓ ) with t i = . . . = t i r = t ∗ + 1 and t i r = . . . = t i l = t ∗ , we get min t ∈ τ t,ℓ,µ n ℓ X i =1 t i o = r · ( t ∗ + 1) + ( ℓ − r ) · t ∗ = t − r ℓ + r. Since max r ∈ N ≤ ℓ − { r − r ℓ } ≤ ℓ it holds that min t ∈ τ t,ℓ,µ n P ℓi =1 t i o ≤ t ℓ + ℓ and hence Vol S ℓ ( t ) ≥ q ( m + η ) t · q − t ℓ − ℓ · γ − ℓq . Since the volumeof a ball is always greater than the volume of a sphere withthe same radius, the statement follows.
Remark 1.
It can be seen from the proof of Lemma 2 that for ℓ | t , we have Vol S ℓ ( t ) ≥ q ( m + η − tℓ ) t · γ − ℓq , i.e., we can drop the term − ℓ in the exponent of q . Theorem 4 (Simplified SP Bound) . For a linear sum-rankmetric code C ( n, k, d ) , the parameters fulfill q mk · q ( m + η − ℓ ⌊ d − ⌋ ) ⌊ d − ⌋− ℓ · γ − ℓq ≥ q mn . Proof:
This follows directly fom Theorem 2 andLemma 2.
Theorem 5 (Asymptotic SP Bound) . Let C ( n, k, d ) be a linearsum-rank metric code and δ := dn the relative minimumdistance. Then the code rate R = kn is upper bounded by R < δ η m − δ (cid:16)
12 + ηm (cid:16)
12 + 1 n (cid:17)(cid:17) + 1 n (cid:16) ηm + ηnm (cid:17) + 1 ηm (cid:16)
14 + log q ( γ q ) (cid:17) + 1 =: R ∗ ( δ ) . Let ξ > be fixed. Then, (i) For m = ηξ → ∞ we get R ∼ δ ξ − δ (cid:16) ξ (cid:17) + 1 . (ii) For ℓ → ∞ one get R ∼ δ η m − δ (cid:16) ηm (cid:17) + ηm (cid:16) + log q ( γ q ) (cid:17) + 1 . Proof:
We transform the simplified SP bound (cf. Theorem 4) with t := ⌊ d − ⌋ into kn ≤ − (cid:16) m + η − tℓ (cid:17) t − ℓ (cid:16) + log q ( γ q ) (cid:17) mn . With δ = dn using t ≥ d − = δn − , it follows kn ≤ δ η m − δ (cid:16)
12 + ηm (cid:16)
12 + 1 n (cid:17)(cid:17) + 1 n (cid:16) ηm + ηnm (cid:17) + 1 ηm (cid:16)
14 + log q ( γ q ) (cid:17) + 1 =: R ∗ ( δ ) . Let ξ be a constant. Consider the following limits:(i) lim m = ηξ →∞ R ∗ ( δ ) = δ ξ − δ (cid:16) ξ (cid:17) + 1 (ii) lim ℓ →∞ R ∗ ( δ ) = δ η m − δ (cid:16) ηm (cid:17) + ηm (cid:16) + log q ( γ q ) (cid:17) + 1 . In a similar fashion, we derive a simplified GV bound, forwhich we rely on an upper bound on
Vol S ℓ ( t ) , which wasderived in [22]. We assume d > to avoid a more technicalstatement. Theorem 6 (Simplified GV Bound) . Let F q m be a finite field, ℓ, n, k, d be positive integers with < d ≤ µℓ that satisfy q m ( k − · ( d − (cid:18) ℓ + d − ℓ − (cid:19) γ ℓq q ( d − m + η − d − ℓ ) < q mn . Then, there is a linear code of length n , dimension k , andminimum ℓ -sum-rank distance at least d .Proof: In [22, Theorem 4] the following upper bound onthe sphere size
Vol S ℓ ( t ) was given: Vol S ℓ ( t ) ≤ (cid:18) ℓ + t − ℓ − (cid:19) γ ℓq q t ( m + η − tℓ ) , ue to Vol B ℓ ( t ) = P tt ′ =0 Vol S ℓ ( t ′ ) ≤ t Vol S ℓ ( t ) fot t > , thisgives an upper bound on Vol B ℓ ( t ) . Together with Theorem 3,the claim follows. Theorem 7 (Asymptotic Gilbert–Varshamov-like Bound) . Fora finite field F q m and positive integers ℓ, n, R n, d with δ := dn and < d ≤ µℓ satisfying R ≤ δ ηm − δ (cid:16) ηm + 2 ηnm (cid:17) + 1 + 1 n + ηnm + ηn m − P δn − i =1 log q (cid:16) ℓ − i (cid:17) + log q ( δn − mn − log q ( γ q ) ηm there exist a linear ℓ -sum-rank metric code of rate R andrelative minimum sum-rank distance at least δ . Let ξ be aconstant. For m = ηξ → ∞ and m ∈ ω (log q ( ℓ )) we have R ∼ δ ξ − δ (cid:16) ξ (cid:17) + 1 Proof:
From the simplified GV bound given in Theorem 6it follows that if q mk ( d − (cid:18) ℓ + d − ℓ − (cid:19) γ ℓq q ( d − m + η − d − ℓ ) ≤ q mn , (4)then there is a ℓ -sum-rank metric code C ( n, k, d ) . We transform4 into kn < (cid:16) dn (cid:17) ηm − dn (cid:16) ηm + 2 ηnm (cid:17) + 1 + 1 n + ηnm + ηn m − log q (cid:16) ( d − · (cid:0) ℓ + d − ℓ − (cid:1)(cid:17) mn − log q ( γ q ) ηm . By substituting δ := dn we get = δ ηm − δ (cid:16) ηm + 2 ηnm (cid:17) + 1 + 1 n + ηnm + ηn m − P δn − i =1 log q (cid:16) ℓ − i (cid:17) + log q ( δn − mn − log q ( γ q ) ηm =: R ∗ . Let ξ be a constant. Consider the following limit: lim m = ηξ →∞ m ∈ ω (log q ( ℓ )) R ∗ = δ ξ − δ (cid:16) ξ (cid:17) + 1 . D. Numerical Comparison
We compare our simplified and asymptotic bounds to theexact bounds in two parameter regimes/asymptotic settings: • Bounded Block Size : We keep the extension degree m andthe block size η constant, and let the number of blocks ℓ go to infinity. This is the case for which there are alreadyasymptotic bounds, see [13]. • Growing Block Size : We let all parameters η, m, ℓ growto infinity proportionally. For the plots with finite param-eters, we choose ℓη and ηm to be constants close to .In contrast to [13], we are able to compare all the bounds forquite large parameters ( n = 2 and even more) since we usethe efficient algorithms for computing the sum-rank ball sizefrom [22].
1) Bounded Block Size:
In Figure 1, we compare oursimplified bounds (cf. Lemma 2 and 6) with the correspondingexact bounds (cf. Theorem 2 and 3) and for the SP boundadditionally with the asymptotic bound given in Theorem 5(ii). Moreover we compare our SP and GV bounds to theasymptotic induced Hamming bound and the asymptotic SPand sphere-covering bound, given in [13, Theorem 4.4 andCorollary 4.10]. The simplified bounds are further away fromthe exact bounds for this parameter regime (compared to the“growing block size” case), since the bounds on
Vol B ℓ ( t ) arebetter suited for ℓ ≈ η . For n = 2 the simplified and theasymptotic SP bounds are nearly identical. From δ ≥ . theasymptotic SP bound is closer to the exact bound than theinduced Hamming bound for this setting of parameters. Theasymptotic sphere-packing and sphere-covering bound [13,Corollary 4.10] nearly match the exact SP and GV bound (for n = 2 ), respectively. . . . . . . . . δ R exact SP ( n = 128 )Simplified SP ( n = 128 )exact SP ( n = 2048 )Simplified SP ( n = 2048 )asymptotic SP ℓ → ∞ induced Hamming bound [13]asymptotic SP [13]exact GV ( n = 128 )Simplified GV ( n = 128 )exact GV ( n = 2048 )Simplified GV ( n = 2048 )asymptotic sphere-covering [13] Fig. 1. comparison of different bounds for fixed value q = 2 η = 8 m = 16 for different values of n ( ℓ = nη )
2) Growing Block Size:
In Figure 2, we compare theasymptotic bounds given in Theorem 5 (i) and Theorem 7with the corresponding exact bounds (cf. Theorem 2 and 3)and with the simplified bounds (cf. Lemma 2 and 6) for twodifferent parameter sets ℓ , m , η and n . One can see that forthis asymptotic setting the simplified and the exact boundsmove closer together for growing η , ℓ . For n = 2 thereis no significant difference between the simplified and theexact bounds. Furthermore, the bounds for this finite n almostcoincide with the asymptotic bounds.IV. G ENERICITY R ESULTS
In this section, we derive the two genericity results. We startwith a statement, that codes attaining almost the GV bound arethe generic case.
A. Random Linear Codes almost attain the GV bound withhigh probability
Theorem 8.
For q, m, n, d , choose ǫ ∈ (cid:16) , − log q (cid:0) Vol B ℓ ( d − mn (cid:1) − n − i and k := n (1 − log q (Vol B ℓ ( d − mn ) − ǫ ) . Let C be chosen uniformly at random from the set of linear codeslength n and dimension k over F q m . Then, C has minimumdistance ≥ d with probability at least − e − Ω( mn ) .Proof: Instead of drawing a code uniformly at randomfrom the set of codes with dimension exactly k , we consider . . . . . . . . δ R exact SP ( n = 64 )Simplified SP ( n = 64 )exact SP ( n = 1024 )Simplified SP ( n = 1024 )asymtotic SP m, η, ℓ → ∞ exact GV ( n = 64 )Simplified GV ( n = 64 )exact GV ( n = 1024 )Simplified GV ( n = 1024 )asymtotic GV m, η, ℓ → ∞ Fig. 2. comparison of different bounds for fixed value of q = 2 and differentvalues of n with η = ℓ and m = 2 · η the following random choice: Choose a matrix G ∈ F k × nq m bydrawing its entries independently uniformly at random, andtake its row space as the code C . Let A be the event that rk( G ) = k and B be the event that the minimum sum-rankdistance of the code is ≥ d . The sought probability of the claimis then given by the conditional probability P ( B | A ) , sincethe event A corresponds to all codes of dimension exactly k .First note that P ( ¬ A ) = P x ∈ F kqm \{ } P ( x · G = ) = P x ∈ F kqm \{ } q mn = ( q mk − · q mn < e − Ω( mn ) . We also bound the probability P ( B ) . For a given i ∈ F kq m ,denote by E i the event that wt SR,ℓ ( i · G ) < d . Then, by theunion bound, we have P ( ¬ B ) = P ( S i ∈ F kqm E i ) ≤ P i ∈ F kqm P ( E i ) . For i = , we have P ( E i ) = 0 and for i = 0 , we get P ( E i ) = P c ∈ F nqm wt SR,ℓ ( c ) ≤ d − P ( i · G = c )= P c ∈ F nqm wt SR,ℓ ( c ) ≤ d − q m · n = Vol B ℓ ( d − q m · n . Hence, we can bound P ( B ) ≥ − q m · k · Vol B ℓ ( d − q m · n ≥ − q − mnǫ for the given choice of k and ǫ . The union bound implies P ( B | A ) ≥ P ( B ∩ A ) ≥ − e − Ω( mn ) , which proves the claim. B. Probability that Random codes are MSRD
In the following, we derive two lower bounds on theprobability that a random linear code is MSRD. The twobounds are adaptions of the two bounds given by Neri et al. in[21] for the rank metric ( ℓ = 1 ), to the general case. As in [21],we use the Schwartz–Zippel Lemma together with a countingargument on matrices. The difference to Neri et al.’s proof isthat these matrices have a special block structure, which resultsin bounds that interpolate smoothly between the Hamming andrank case. It is interesting to note that, in contrast to the twobounds in [21], the two bounds are advantageous over the otherin different parameter ranges. This is due to the nature of theused bounds on the number of these matrices.Recall that the Schwartz–Zippel Lemma states that, for anon-zero polynomial f ∈ F q m [ x , . . . , x r ] of degree d ≥ and independently uniformly distributed random variables v , . . . , v r over a subset F of F q m the following probabilitybound holds: P r ( f ( v , . . . , v r ) = 0) ≤ d |F| .We start with a characterization of a code being MSRD(Lemma 3 below) and use the following notation. Notation 1.
We denote by A ℓ,t and U ℓ,t the following sets ofblock matrices. A ℓ,t := n A = M ℓi =1 A i ∈ F t × nq | A i ∈ F t i × ηq , rk( A i ) = t i , P ℓi =0 t i = t o , ∀ t ∈ { , . . . , n } . U ℓ,t := n U = ℓ M i =1 U i ∈ A ℓ,t | U i upper triangular matrix o . In the following lemma, the equivalence (i) ⇔ (ii) was alreadystudied in a similar form in [14], [22], [26]. Lemma 3.
Let C ( n, k, d ) be a linear sum-rank metric codewith parity check matrix H ∈ F n − k × nq m and generator matrix G ∈ F k × nq m . The following statements are equivalent: (i) C is MSRD (ii) rk F qm ( AG ⊤ ) = k ∀ A ∈ A ℓ,k (iii) rk F qm ( U G ⊤ ) = k ∀ U ∈ U ℓ,k .Proof: The equivalence (i) ⇔ (ii) follows directly from[14], [22], [26]. For (ii) ⇔ (iii), since U ℓ,k ⊂ A ℓ,k it issufficient to show, that (iii) ⇒ (ii). Therefore we assume, that rk F qm ( U G ⊤ ) = k ∀ U ∈ U ℓ,k . Let A = L ℓi =1 A i ∈ A ℓ,k .Then A i ∈ F t i × ηq with rk( A i ) = t i ∀ i ∈ { , . . . , ℓ } and P ℓi =0 t i = t . For each i ∈ { , . . . , ℓ } let U i be thereduced echolon form of A i , then there is a regular matrix X i ∈ F t i × t i q with A i = X i U i . We define X := L ℓi =1 X i and U := L ℓi =1 U i , i.e. X is regular, U ∈ U ℓ,k and A = XU . Since X i has full rank, it holds rk F qm ( AG ⊤ ) =rk F qm ( XU G ⊤ ) = rk F qm ( U G ⊤ ) = k. We also derive upper bounds on the cardinality |A ℓ,t | and |U ℓ,t | . Lemma 4.
For the cardinality of A ℓ,t it holds |A ℓ,t | = X t ∈ τ t,ℓ,µ Y t i − j =0 ( q η − q j ) ≤ (cid:0) t + ℓ − ℓ − (cid:1) q ηt . Proof:
Since the number of matrices A i ∈ F η × t i q of rank t i is Y t i − j =0 ( q η − q j ) ≤ q ηt i ∀ i = 1 , . . . , ℓ, it follows, that for a fixed weight decomposition t = P ℓi =1 t i with the restriction, ≤ t i ≤ η the number of matrices A = A ⊕ A ⊕ . . . ⊕ A ℓ ∈ F n × tq with A i ∈ F η × t i q and rk( A i ) = t i is Q ℓi =1 Q t i − j =0 ( q η − q j ) ≤ q η P ℓi =1 t i = q ηt . With the numberof ordered partitions τ t,ℓ,µ and its upper bound (see (1)) weget |A ℓ,t | = P t ∈ τ t,ℓ,µ Q ℓi =1 Q t i − j =0 ( q η − q j ) ≤ (cid:0) t + ℓ − ℓ − (cid:1) q ηt . Lemma 5.
For the cardinality of U ℓ,t it holds |U ℓ,t | = P t ∈ τ t,ℓ,µ Q ℓi =1 (cid:20) ηt i (cid:21) q ≤ (cid:0) t + ℓ − ℓ − (cid:1) q t ( η − tℓ ) · γ ℓq . Proof:
The number of upper triangular matrices U i ∈ F t i × ηq of rank t i is equal to the number of t i -dimensional sub-spaces of F ηq and therefore equal to the q -binomial coefficient ηt i (cid:21) q . With the same arguments as in the proof of Lemma 4the equality |U ℓ,t | = P t ∈ τ t,ℓ,µ Q ℓi =1 (cid:20) ηt i (cid:21) q follows. Using theupper bound (cid:20) ηt i (cid:21) q < γ q · q t i ( η − t i ) (see [25], [27]) we get |U ℓ,t | < P t ∈ τ t,ℓ,µ γ ℓq q P ℓi =1 t i η − t i ≤ | τ t,ℓ,µ | · γ ℓq q tη · max t ∈ τ t,ℓ,µ q − P ℓi =1 t i . As derived in [22, Proof of Theorem 4] the choice t i = tℓ leadsto the upper bound max t ∈ τ t,ℓ,µ − n P ℓi =1 t i o ≤ − t ℓ . Hence |U ℓ,t | < | τ t,ℓ,µ | · γ ℓq q tη − t ℓ . With the upper bound on the cardinality of τ t,ℓ,µ (see (1)) thestatement follows.The lemmas above give the following probability bound.The bound converges to for fixed n, k, ℓ and m → ∞ . Thismeans that for large enough extension degree m , most linearcodes are MSRD. Theorem 9.
Let G be a systematic generator matrix [ I k | X ] ∈ F k × nq m , where entries of X are independently anduniformly chosen from F q m , and denote by C the row spaceof G . Then, C is an MSRD code (w.r.t. ℓ ) with probability atleast − k (cid:0) k + ℓ − ℓ − (cid:1) q ηk − m . Proof:
From Lemma 3 we know that C is MSRD if andonly if rk F qm ( G · A ) = k for all A ∈ A ℓ,k . This leads to thefact that C is not MSRD if and only if there exists a matrix A ∈ A ℓ,k , such that det( G · A ) = 0 . Considering the entriesof X as variables x , . . . x k ( n − k ) ,then it holds for the product G · A = [ I k | X ] · A = (cid:16) a ij + P nr = k +1 x ir − k a rj (cid:17) ij that each variable x , . . . x k ( n − k ) is contained in at most onerow. Hence, the determinant of this product is a multivariatepolynomial f A := det([ I k | X ] · A ) ∈ F q [ x , . . . x k ( n − k ) ] ofdegree at most k for each A ∈ A ℓ,k . Using the notation f ( x , . . . x k ( n − k ) ) := Q A ∈A ℓ,k f A ( x , . . . x k ( n − k ) ) we have P ( C is not MSRD ) = P ( f ( x , . . . x k ( n − k ) ) = 0) .Since f = 0 and the variables x , . . . x k ( n − k ) are indepen-dently and uniformly distributed, it follows with the Schwartz–Zippel Lemma that the probability P ( f ( x , . . . x k ( n − k ) ) =0) ≤ deg f | F qm | = k |A ℓ,k | q m . From Lemma 4 it follows |A ℓ,k | ≤ (cid:0) k + ℓ − ℓ − (cid:1) q ηk , which proves the claim.The number of matrices in U ℓ,t is always smaller thanthe number of matrices in A ℓ,t . This motivates the followingbound, which uses the criterion of Lemma 3 (iii) and Lemma 5. Theorem 10.
Let G = [ I k | X ] ∈ F k × nq m be a systematicgenerator matrix, with a random matrix X , i.e., the entriesare independently and uniformly chosen from F q m . The rowspace of G is a is a linear ℓ -sum–rank metric code over F q m ,which we denote by C . The probability p that C is MSRD islower bounded by p ≥ − k (cid:0) k + ℓ − ℓ − (cid:1) q k ( η − kℓ ) − ℓ − m · γ ℓq . Proof:
With Lemma 3 (iii) and Lemma 5 using the samearguments as in the proof of Theorem 9 by just replacing A ℓ,k by U ℓ,k one gets P ( C is not MSRD ) ≤ k |U ℓ,k | q m ≤ k (cid:0) k + ℓ − ℓ − (cid:1) q k ( η − kℓ ) − ℓ · γ ℓq q − m . C. Numerical Comparison
For the bounds given in Theorem 9 and 10, Figure 3 showsthe minimal values of the extension degree m , for which thereis a non-zero probability that a code is MSRD for varyingnumbers of blocks ℓ and constant values of n , k and q . Theseminimal values m min smoothly interpolate between the knownextreme cases: ℓ = 1 (cf. [21]) and ℓ = n (cf. [20]). Sincethe complementary probability of the two bounds differ inthe factor ζ := q − k ℓ − ℓ γ ℓq it depends on the relation of ζ to , which of the two bounds is the better one. This can beobserved in Figure 3. For small values of ℓ , the bound derivedin Theorem 10 is better and in turn for large values of ℓ ,Theorem 9 provides the better bound. Hence, both bounds haveadvantages in certain parameter ranges of ℓ . This is differentfrom the bounds in [21], where the bound in [21, Theorem 26]is an improvement of the bound derived in [21, Theorem 21]. ℓ m m i n Theorem 9Theorem 10
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