Decoding of Interleaved Linearized Reed-Solomon Codes with Applications to Network Coding
aa r X i v : . [ c s . I T ] J a n Decoding of Interleaved Linearized Reed–SolomonCodes with Applications to Network Coding
Hannes Bartz
Institute of Communications and NavigationGerman Aerospace Center (DLR), Germany [email protected]
Sven Puchinger
Department of Applied Mathematics and Computer ScienceTechnical University of Denmark (DTU), Denmark [email protected]
Abstract —Recently, Mart´ınez-Pe˜nas and Kschischang (IEEETrans. Inf. Theory, 2019) showed that lifted linearized Reed–Solomon codes are suitable codes for error control in multi-shot network coding. We show how to construct and decodelifted interleaved linearized Reed–Solomon codes. Compared to theconstruction by Mart´ınez-Pe˜nas–Kschischang, interleaving allowsto increase the decoding region significantly (especially w.r.t. thenumber of insertions) and decreases the overhead due to the lifting(i.e., increases the code rate), at the cost of an increased packet size.The proposed decoder is a list decoder that can also be interpretedas a probabilistic unique decoder. Although our best upper boundon the list size is exponential, we present a heuristic argument andsimulation results that indicate that the list size is in fact one formost channel realizations up to the maximal decoding radius.
Index Terms —Multishot network coding, subspace codes, sum-rank metric, multishot operator channel
I. I
NTRODUCTION
Network coding [1] is a powerful approach to achieve thecapacity of multicast networks. Unlike the classical routingschemes, network coding allows to mix (e.g. linearly combine)incoming packets at intermediate nodes. K¨otter and Kschis-chang proposed codes in the subpsace metric as a suitable toolfor error correction in (random) linear network coding [2].The sum-rank metric is a hybrid between the Hammingand the rank metric. Already in [3], codes in this metric(called “extended rank metric” therein) were proposed for errorcorrection in multishot network coding. Since then, there havebeen many works on code constructions and efficient decodingalgorithms for codes in the sum-rank metric, including [4]–[15].The first general class of codes attaining the Singleton bound inthe sum-rank metric is called linearized Reed–Solomon (LRS)codes [9], which can be seen as a mix of Reed–Solomon codes(Hamming metric) and Gabidulin codes (rank metric) [16]–[18].It was shown in [11] that lifted
LRS codes provide reliable andsecure coding schemes for non-coherent network coding underan adversarial model.An s -interleaved code is a direct sum of s codes of thesame length (called constituent codes). This means that if theconstituent codes are over F q , then the interleaved code canbe viewed as a (not necessarily linear) code over F q s . In theHamming and rank metric, there are various decoders that cansignificantly increase the decoding radius of a constitutent codeby collaboratively decoding in an interleaved variant thereof.Such decoders are known in the Hamming metric for Reed–Solomon [19]–[31] and in general algebraic geometry codes[32]–[34], and in the rank metric for Gabidulin codes [13], [35]–[41]. All of these decoders have in common that they are either Sven Puchinger has received funding from the European Union’s Horizon2020 research and innovation program under the Marie Sklodowska-Curie grantagreement no. 713683. list decoders with exponential worst-case and small average-case list size, or probabilistic unique decoders that fail with avery small probability.Interleaving was suggested in [42] as a method to decrease theoverhead in lifted Gabidulin codes for error correction in non-coherent (single-shot) network coding, at the cost of a largerpacket size while preserving a low decoding complexity. It waslater shown [37], [39], [43] that it can also increase the error-correction capability of the code using suitable decoders forinterleaved Gabidulin codes.
A. Our Contributions
In this paper, we define lifted interleaved linearized Reed–Solomon codes and propose a novel interpolation-based listdecoder that is based on the list decoder by Wachter-Zeh andZeh [39] for interleaved Gabidulin codes. We derive a decodingregion for the codes in the sum-subspace metric, analyze thecomplexity of the decoder, give an exponential upper bound onthe list size, and give heuristic arguments and numerical evi-dence that the output list size is with overwhelming probabilityequal to one for random realizations of the multishot operatorchannel that stay within the decoding region.Compared to [11], we decrease the relative overhead intro-duced by lifting (or equivalently, increase the rate for the samecode length and block size) and at the same time extend thedecoding region significantly, especially the resilience againstinsertions in the multishot operator channel. These advantagescome at the cost of a larger packet size of the packets within thenetwork and a supposedly small failure probability. It is workin progress to derive a formal bound on the failure probability.Moreover, for the case s = 1 (no interleaving), our algorithmdoes not require the assumption from [11, Sec. V.H] that n r ≤ n t , which means that our decoder works in cases in which [11]does not work. II. P RELIMINARIES
Let F q be a finite field of order q and denote by F q m theextension field of F q of degree m with primitive element α .The multiplicative group F q m \ { } of F q m is denoted by F ∗ q m . Matrices and vectors are denoted by bold uppercase andlowercase letters like A and a , respectively. Under a fixed basisof F q m over F q any element a ∈ F q m can be represented bya corresponding vector a ∈ F mq . For A ∈ F M × Nq m we denoteby h A i q the F q -linear rowspace of the matrix A q ∈ F M × Nmq obtained by row-wise expanding the elements in A over F q . . Skew Polynomials Let σ : F q m → F q m be a finite field automorphism. A skewpolynomial is a polynomial of the form f ( x ) = P i f i x i (1)with a finite number of coefficients f i ∈ F q m being nonzero.The degree deg( f ) of a skew polynomial f is defined as max { i : f i = 0 } if f = 0 and −∞ otherwise.The set of skew polynomials with coefficients in F q m togetherwith ordinary polynomial addition and the multiplication rule xa = σ ( a ) x, a ∈ F q m (2)forms a non-commutative ring denoted by F q m [ x, σ ] . The set ofskew polynomials in F q m [ x, σ ] of degree less than k is denotedby F q m [ x, σ ] For two skew polynomials f, g ∈ F q m [ x, σ ] and elements a, b ∈ F q m the generalized operatorevaluation of the product f · g at b w.r.t a is given by ( f · g )( b ) a = f ( g ( b ) a ) a . (5) Proof: We have that ( f · g )( b ) a = P j f j (cid:0)P i σ j ( g i − j ) σ i ( b ) N i ( a ) (cid:1) . By defining κ = i − j we get ( f · g )( b ) a = P j f j σ j (cid:0)P κ g κ σ j ( b ) σ − j ( N κ + j ( a )) (cid:1) . Due to σ − j ( N κ + j ( a )) = N κ ( a ) σ − j ( N j ( a )) we have that ( f · g )( b ) a = P j f j σ j (cid:0)P κ g κ σ j ( b ) σ − j ( N κ + j ( a )) (cid:1) = P j f j D ja ( P κ g κ D κa ( b )) = f ( g ( b ) a ) a . B. Conjugacy Classes Two elements a, b ∈ F q m are called conjugates if there existsan element c ∈ F ∗ q m s.t. a c := σ ( c ) ac − . The set C ( a ) := (cid:8) a c : c ∈ F ∗ q m (cid:9) (6)is called conjugacy class of a . A finite field F q m has at most ℓ ≤ q − distinct conjugacy classes. For ℓ ≤ q − the elements , α, α , . . . , α ℓ − are representatives of all (nontrivial) disjointconjugacy classes of F q m . Proposition 1 (Number of Roots [12]) Let β ( i )1 , . . . , β ( i ) n i beelements from F q m and let a , . . . , a ℓ be representatives be fromconjugacy classes of F q m for all i = 1 , . . . , ℓ . Then for anynonzero f ∈ F q m [ x, σ ] satisfying f ( β ( i ) j ) a i = 0 , ∀ i = 1 , . . . , ℓ, j = 1 , . . . , n i (7) we have that deg( f ) ≤ P ℓi =1 n i where equality holds iff the β ( i )1 , . . . , β ( i ) n i are F q -linearly independent for each i = 1 , . . . , ℓ . C. Multi-Shot Network Coding As a channel model we consider the multishot operator chan-nel from [3] which consists of multiple independent channeluses of the operator channel from [2]. Let P q ( N i ) denote theset of all subspaces of F N i q . For N = N + N + · · · + N ℓ wedefine the ℓ -fold Cartesian product P q ( N ) := Q ℓi =1 P q ( N i ) = P q ( N ) × · · · × P q ( N ℓ ) . (8)The Grassmannian, i.e. the set of all subspaces of dimension l in P q ( N i ) , is denoted by G q ( N i , l ) .We now consider ℓ independent channel uses of the operatorchannel [2]. We consider sum-constant-dimension codes, i.e.codes that inject the same number of (linearly independent)packets n ( i ) t in a given shot. In the i -th shot, the operator channeltakes a subspace V i ∈ G q ( N i , n ( i ) t ) and returns a subspace U i = H n ( i ) t − δ ( i ) ( V i ) ⊕ E i (9)where H n ( i ) t − δ ( i ) returns a random ( n ( i ) t − δ ( i ) ) -dimensionalsubspace of V i and E i is an error subspace of dimension γ ( i ) =dim( E i ) with V i ∩ E i = . Hence, each received subspace U i has dimension n ( i ) r := dim( U i ) = n ( i ) t + γ ( i ) − δ ( i ) . The overalltransmitted/received words are tuples of subspaces V = ( V , V , . . . , V ℓ ) ∈ Q ℓi =1 G q ( N i , n ( i ) t ) , (10) U = ( U , U , . . . , U ℓ ) ∈ Q ℓi =1 G q ( N i , n ( i ) r ) , (11)where we define n t = P ℓi =1 n ( i ) t and n r = P ℓi =1 n ( i ) r . After ℓ channel uses (or shots ) we have that n r = P ℓi =1 n ( i ) r = P ℓi =1 n ( i ) t + γ ( i ) − δ ( i ) . (12)Setting γ = P ℓi =1 γ ( i ) , δ = P ℓi =1 δ ( i ) we may write (12) as n r = n t + γ − δ. (13) Definition 1 (Sum-Subspace Distance [3]) Given U = ( U , U , . . . , U ℓ ) and V = ( V , V , . . . , V ℓ ) ∈ P q ( N ) thesum-subspace distance between U and V is defined as d Σ S ( U , V ) := P ℓi =1 dim( U i + V i ) − dim( U i ∩ V i ) . (14)Similar to [45] we define the ( γ, δ ) reachability for multipleindependent operator channel uses. Definition 2 ( ( γ, δ ) Reachability) Given two tuples of sub-spaces U = ( U , U , . . . , U ℓ ) and V = ( V , V , . . . , V ℓ ) ∈P q ( N ) we say that V is ( γ, δ ) -reachable from U if thereexists a realization of the multishot operator channel (9) with γ = P ℓi =1 γ ( i ) insertions and δ = P ℓi =1 δ ( i ) deletions thattransforms the input V to the output U . Proposition 2 Consider U = ( U , U , . . . , U ℓ ) and V =( V , V , . . . , V ℓ ) ∈ P q ( N ) . If V is ( γ, δ ) -reachable from U ,then we have that d Σ S ( U , V ) = γ + δ . III. L IFTED I NTERLEAVED L INEARIZED R EED –S OLOMON C ODES In this section we consider lifted interleaved linearized Reed–Solomon (LILRS) codes for multiple transmissions over theoperator channel. We generalize the ideas from [11] to obtainmultishot subspace codes by lifting interleaved linearized Reed–Solomon (ILRS) codes. efinition 3 (Lifted Interleaved Linearized RS Code) Let α be a primitive element of F q m . Let β ( i )1 , . . . , β ( i ) n ( i ) t be F q -linearly independent elements from F q m and definethe vectors β ( i ) = ( β ( i )1 , β ( i )2 , . . . , β ( i ) n ( i ) t ) ∈ F n ( i ) t q m for all i = 1 , . . . , ℓ . Further, define β = (cid:0) β (1) , β (2) , . . . , β ( ℓ ) (cid:1) ∈ F n t q m .A lifted s -interleaved linearized Reed–Solomon (LILRS)code LILRS[ β , ℓ, s ; n t , k ] of subspace dimension n t = n (1) t + n (2) t + · · · + n ( ℓ ) t and dimension k ≤ n t isdefined as n V ( f ) := ( V ( f ) , . . . , V ℓ ( f )) : f ∈ F q m [ x, σ ] s LILRS[ β , ℓ, s ; n t , k ] isdefined as R = log q ( | LILRS[ β , ℓ, s ; n t , k ] | ) P ℓi =1 n ( i ) t N i = smk P ℓi =1 n ( i ) t ( n ( i ) t + sm ) . (15)Note, that there exist other definitions of the code rate formultishot codes, which are discussed in [3, Section IV.A].The definition of LILRS codes generalizes several codefamilies. For s = 1 we obtain the lifted linearized Reed–Solomon codes from [11, Section V.III]. For ℓ = 1 we obtainlifted interleaved Gabidulin codes as considered in e.g. [43],[46] with K¨otter–Kschischang codes [2] as special case for s = 1 . Without lifting we obtain interleaved linearized Reed–Solomon codes with linearized Reed–Solomon codes [9] asspecial case for s = 1 . Proposition 3 The minimum sum-subspace distance of a LILRScode LILRS[ β , ℓ, s ; n t , k ] as in Definition 3 is d Σ S (LILRS[ β , ℓ, s ; n t , k ]) = 2 ( n t − k + 1) . (16)IV. A N I NTERPOLATION -B ASED D ECODING A PPROACH We now derive an interpolation-based decoding approachfor LILRS codes. The decoding principle consists of an inter-polation step and a root-finding step. In [11], (lifted) linearizedReed–Solomon codes are decoded using the isometry betweenthe sum-rank and the skew metric. In this paper we consider aninterpolation-based decoding scheme in the generalized operatorevaluation domain without the need for casting the decodingproblem to the skew metric. The new decoder is a generalizationof [39] (interleaved Gabidulin codes in the rank metric) and [43](lifted interleaved Gabidulin codes in the subspace metric). A. Interpolation Step Suppose we transmit the tuple of subspaces V ( f ) = ( V ( f ) , . . . , V ℓ ( f )) ∈ Q ℓi =1 G q ( N i , n ( i ) t ) (17)over the multishot operator channel and receive the subspaces U = ( U , . . . , U ℓ ) ∈ Q ℓi =1 G q ( N i , n ( i ) r ) , (18) where, for all i = 1 , . . . , ℓ , U i = * ξ ( i )1 u (1 ,i )1 u (2 ,i )1 . . . u ( s,i )1 ... ... ... . . . ... ξ ( i ) n ( i ) r u (1 ,i ) n ( i ) r u (2 ,i ) n ( i ) r . . . u ( s,i ) n ( i ) r + q . (19) Remark 1 In contrast to [11, Section V.III] we do not need theassumption that the dimension(s) of the transmitted subspace(s)equals the dimension(s) of the received subspace(s). For a multivariate skew polynomial of the form Q ( x, y , . . . , y s ) = Q ( x ) + Q ( x ) y + · · · + Q s ( x ) y s (20)where Q l ( x ) ∈ F q m [ x, σ ] for all l ∈ [0 , s ] define the n r evaluations E ( i ) j for j = 1 , . . . , n ( i ) r and i = 1 , . . . , ℓ as E ( i ) j ( Q ) := Q ( ξ ( i ) j ) α i − + P sl =1 Q l ( u ( l,i ) j ) α i − . (21)The w -weighted degree of a multivariate skew polynomial Q as in (20) is defined as deg w ( Q ) = max j { deg( Q j )+ w j } . Nowconsider the following interpolation problem in F q m [ x, σ ] . Problem 1 (Generalized Operator Interpolation Problem) Given the integers D, s ∈ Z + , the set of points P = { ( ξ (1)1 , u (1 , , . . . ,u ( s, ) , ( ξ (1)2 , u (1 , , . . . , u ( s, ) . . .. . . , ( ξ ( ℓ ) n ( ℓ ) r , u (1 ,ℓ ) n ( ℓ ) r , . . . , u ( s,ℓ ) n ( ℓ ) r ) } ⊂ F s +1 q m and a vector w ∈ Z s +1+ , find a nonzero polynomial of the form Q ( x, y , . . . , y s ) = Q ( x ) + Q ( x ) y + · · · + Q s ( x ) y s (22) with Q l ( x ) ∈ F q m [ x, σ ] for all l ∈ [0 , s ] that satisfies: E ( i ) j ( Q ) = 0 , ∀ i = 1 , . . . , ℓ , j = 1 , . . . , n ( i ) r , deg w ( Q ( x, y , . . . , y s )) < D . A solution of Problem 1 can be found by calling the skewK¨otter interpolation [47] with evaluation maps E ( i ) j as definedin (21) requiring O (cid:0) s n (cid:1) operations in F q m . Lemma 2 (Existence of Solution) A nonzero solution ofProblem 1 exists if D = (cid:6) n r + s ( k − s +1 (cid:7) .Proof: Problem 1 corresponds to a system of n r F q m -linearequations in D ( s +1) − s ( k − unknowns which has a nonzerosolution the number of equations is less than the number ofunknowns, i.e. if n r < D ( s + 1) − s ( k − ⇐⇒ D ≥ n r + s ( k − s +1 . (23) B. Root-Finding Step The goal of the root-finding step is to recover the messagepolynomials f (1) , . . . , f ( s ) ∈ F q m [ x, σ ] Let P ( x ) := Q ( x ) + Q ( x ) f (1) ( x ) + · · · + Q s ( x ) f ( s ) ( x ) . (24) Then there exist elements ζ ( i )1 , . . . , ζ n ( i ) t − δ ( i ) in F q m that are F q -linearly independent for each i = 1 , . . . , ℓ such that P ( ζ ( i ) j ) α i − = 0 (25) or all i = 1 , . . . , ℓ and j = 1 , . . . , n ( i ) t − δ ( i ) .Proof: In each shot the noncorrupted intersection space hasdimension dim( U i ∩ V i ) = n ( i ) t − δ ( i ) for all i = 1 , . . . , ℓ . Abasis for each intersection space U i ∩ V i can be represented as n(cid:16) ζ ( i ) j , f (1) ( ζ ( i ) j ) α i − ,. . . , f ( s ) ( ζ ( i ) j ) α i − (cid:17) : j ∈ [1 , n ( i ) t − δ ( i ) ] o (26)where ζ ( i )1 , . . . , ζ ( i ) n ( i ) t − δ ( i ) are ( n ( i ) t − δ ( i ) ) F q -linearly inde-pendent elements from F q m for all i = 1 , . . . , ℓ . Since eachintersection space U i ∩ V i is a subspace of the received space U i we have that P ( ζ ( i ) j ) α i − := Q ( ζ ( i ) j ) α i − + s X l =1 Q l ( f ( l ) ( ζ ( i ) j ) α i − ) α i − = 0 (27)for all i = 1 , . . . , ℓ, j = 1 , . . . , n ( i ) t − δ ( i ) . Theorem 1 (Decoding Region) Let U ∈ Q ℓi =1 G q ( N i , n ( i ) r ) be the tuple containing the received subspaces and let Q ( x, y , . . . , y s ) = 0 fulfill the constraints in Problem 1. Thenfor all codewords V ( f ) ∈ LILRS[ β , ℓ, s ; n t , k ] that are ( γ, δ ) -reachable from U , where γ and δ satisfy γ + sδ < s ( n t − k + 1) , (28) we have that P ( x ) = Q ( x ) + Q ( x ) f (1) ( x ) + . . . + Q s ( x ) f ( s ) ( x ) = 0 . (29) Proof: By Lemma 3 there exist elements ζ ( i )1 , . . . , ζ n ( i ) t − δ ( i ) in F q m that are F q -linearly independent for each i = 1 , . . . , ℓ such that P ( ζ ( i ) j ) α i − = 0 (30)for all i = 1 , . . . , ℓ and j = 1 , . . . , n ( i ) t − δ ( i ) . By choosing D ≤ n t − δ (31)the degree of P ( x ) exceeds the degree bound from Proposition 1which is possible only if P ( x ) = 0 . Combining (23) and (31)we get n r + s ( k − < D ( s + 1) ≤ ( s + 1)( n t − δ ) ⇐⇒ γ + sδ < s ( n t − k + 1) . The decoding region in (28) shows and improved insertion-correction performance due to interleaving.In the root-finding step, all polynomials f (1) , . . . , f ( s ) ∈ F q m [ x, σ ] Instead of using only one solution of Problem 1, we followthe ideas of [46] and use a basis of the F q m -linear solutionspace of the interpolation problem in order to derive bounds onthe worst-case and average list size. In [48] it was shown thatusing a degree-restricted subset of a Gr¨obner basis for the left F q m [ x, σ ] -linear interpolation submodule achieves the smallestpossible list size. Let the dimension of the F q m -linear solution space of Problem 1 be d I and define the corresponding basispolynomials as Q ( r )0 ( x ) = P D − i =0 q ( r )0 ,i x i , Q ( r ) j ( x ) = P D − ki =0 q ( r ) j,i x i (32)for all j = 1 , . . . , s and r = 1 , . . . , d I . Define the matrix Q ij = σ i (cid:16) q (1)1 ,j (cid:17) σ i (cid:16) q (1)2 ,j (cid:17) . . . σ i (cid:16) q (1) s,j (cid:17) ... ... . . . ... σ i (cid:16) q ( r )1 ,j (cid:17) σ i (cid:16) q ( r )2 ,j (cid:17) . . . σ i (cid:16) q ( r ) s,j (cid:17) (33)and the vectors f ij := (cid:16) σ i (cid:16) f (1) j (cid:17) , . . . , σ i (cid:16) f ( s ) j (cid:17)(cid:17) (34)and q i ,j := (cid:16) σ i (cid:16) q (1)0 ,j (cid:17) , . . . , σ i (cid:16) q ( r )0 ,j (cid:17)(cid:17) . (35)Defining the root-finding matrix Q = Q Q − Q − ... Q − . . . Q − ( D − k ) D − k ... . . . Q − ( k − . . . . . . Q − k . . . ... Q − ( D − D − k (36)and the vectors f = (cid:16) f , . . . , f − ( k − k − (cid:17) T and q = (cid:16) q , , . . . , q − ( D − ,D − (cid:17) T we can write the root-finding system (29) as Q · f = − q . (37)In general, the root-finding matrix Q in (36) can be rankdeficient. In this case we obtain a list of potential messagepolynomials f (1) , . . . , f ( s ) . Using the same arguments as in [39]on the structure of Q in (36), one can derive a lower boundon the rank of Q , and thus the following upper bound on thenumber of solutions of (37). Lemma 4 (Worst-Case List Size) The root-findingsystem in (29) has at most q m ( k ( s − solutions f (1) , . . . , f ( s ) ∈ F q m [ x, σ ] We now consider the interpolation-based decoder from Sec-tion IV as a probabilistic-unique decoder which either returnsa unique solution (if the list size is equal to one) or a decodingfailure. In order to get an estimate of the decoding failureprobability P f , we use similar assumptions as in [46] to derivea heuristic upper bound.Using similar arguments as in [46, Lemma 3] it can beshown that the dimension d I of the F q m -linear solution spaceof Problem 1 satisfies d I ≥ s ( D + 1) − sk − γ. (38)he rank of the root-finding matrix Q can be full if and only ifthe dimension of the solution space of the interpolation problem d I is at least s , i.e. if d I ≥ s ⇐⇒ γ + sδ ≤ s ( n t − k ) (39)The probabilistic-unique decoding region in (39) is only sightlysmaller than the list decoding region in (28). The improveddecoding region for LILRS codes is illustrated in Figure 1. s ss − s − s − γ insertions δ d e l e ti on s Martinez-Kschischang [11] ( s = 1) Probabilistic unique decoding (39) ( s = 4) List decoding (28) ( s = 4) Fig. 1. Decoding region for Martinez-Kschischang [11] codes ( s = 1) and fordecoding of lifted ( s = 4) -interleaved linearized Reed–Solomon codes. Thedecoding region for insertions increases with the interleaving order s . Combining (31) and (39) we get the degree constraint D = ⌈ n r + sks +1 ⌉ for the probabilistic-unique decoder (see [48]).Under the assumption that the coefficients q ( r ) i,j are uniformlydistributed over F q m (see [46, Lemma 9]) we have that P f ≤ q − m ( d I − s +1) ≤ q − m ( s ( ⌈ nr + sks +1 ⌉ − k ) − γ +1 ) . (40)Our simulations results in the next section indicate that this isalso a good estimate of the failure probabilty for subspace tuplesthat are chosen uniformly at random from the set of subspacesfrom which a given transmitted codeword is ( γ, δ )-reachable.As in [39], [43] for interleaved Gabidulin and lifted interleavedGabidulin codes, respectively, it is an open problem to derivea formal failure probability bound for this non-uniform input.This is subject to future work.V. C OMPARISON TO P REVIOUS W ORK AND S IMULATION R ESULTS The relative overhead due to the lifting is reduced withincreasing interleaving order. Table I shows the improvementof the code rate for increasing interleaving orders. TABLE IC OMPARISON OF THE DIMENSION N i OF THE ELEMENTARY AMBIENTSPACES AND THE CODE RATE R BETWEEN LLRS AND LILRS CODES .LLRS [11] LILRSDimension N i (“packet size”) n ( i ) t + m n ( i ) t + sm Code Rate R mk P ℓi =1 n ( i ) t ( n ( i ) t + m ) smk P ℓi =1 n ( i ) t ( n ( i ) t + sm ) In order to verify the heuristic upper bound on the decodingfailure probability in (40) we performed a Monte Carlo simula-tion ( errors) of a code LILRS[ β , ℓ = 2 , s = 3; n t = 6 , k =3] over F over a multishot operator channel with overall δ = 1 deletion and γ ∈ { , , } insertions.The channel realization is chosen uniformly at random fromall possible realizations of the multishot operator channel withexactly this number of deletions and insertions. We implemented this drawing procedure by adapting the efficient dynamic-programming routine in [49, Remark 7] for drawing an error ofgiven sum-rank weight uniformly at random.The results in Figure 2 show, that the heuristic upper boundin (40) gives a good estimate of the decoding failure probability P f . For the same parameters a (non-interleaved) lifted linearized − − − − − γ + sδ D ec od i ng f a il u r e p r ob a b ilit y P f Upper bound on P f (40)Simulation δ = 1 Fig. 2. Result of a Monte Carlo simulation of the code LILRS[ β , ℓ = 2 , s =3; n t = 6 , k = 3] over F transmitted over a multishot operator channel withoverall δ = 1 deletions and γ = 4 , , insertions. Reed–Solomon code [11] (i.e. s = 1 ) can only correct γ insertions and δ deletions up to γ + δ < .VI. F URTHER A PPLICATIONS The decoding scheme in Section IV can be used to decodeILRS codes (without lifting) in the sum-rank metric and inter-leaved skew Reed–Solomon (ISRS) codes in the skew metric.For the definition of the sum-rank and the skew metric the readeris referred to e.g. [9]).Let the parameters be as in Definition 3 and define n i = n ( i ) t = n ( i ) r for all i = 1 , . . . , ℓ . Then an s -interleavedlinearized Reed–Solomon code ILRS[ β , ℓ, s ; n, k ] of length n and dimension k is defined as f (1) ( β (1) ) . . . f (1) ( β ( ℓ ) ) α ℓ − ... . . . ... f ( s ) ( β (1) ) . . . f ( s ) ( β ( ℓ ) ) α ℓ − : f ( j ) ∈ F q m [ x, σ ] IEEE Transactions on information theory , vol. 46, no. 4, pp. 1204–1216, 2000.[2] R. Koetter and F. R. Kschischang, “Coding for Errors and Erasures inRandom Network Coding,” IEEE Transactions on Information theory ,vol. 54, no. 8, pp. 3579–3591, 2008.[3] R. W. N´obrega and B. F. 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