Theory of supports for linear codes endowed with the sum-rank metric

Abstract

The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (support spaces) they define are obtained; the classical operations of restriction and shortening are extended to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found. Three main applications are given: (1) Generalized sum-rank weights are introduced, together with their basic properties and bounds; (2) It is shown that duals, shortened and restricted codes of maximum sum-rank distance (MSRD) codes are in turn MSRD; (3) Degenerateness and effective lengths of sum-rank codes are introduced and characterized. In an Appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given.

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References

  1. 1.

    Barg A.: The matroid of supports of a linear code. Appl. Algebra Eng. Commun. Comput. 8(2), 165–172 (1997).

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Berger T.P.: Isometries for rank distance and permutation group of Gabidulin codes. IEEE Trans. Inf. Theory 49(11), 3016–3019 (2003).

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Blaum M., Hafner J.L., Hetzler S.: Partial-MDS codes and their application to RAID type of architectures. IEEE Trans. Inf. Theory 59(7), 4510–4519 (2013).

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Boucher D., Ulmer F.: Linear codes using skew polynomials with automorphisms and derivations. Des. Codes Cryptogr. 70(3), 405–431 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Byrne E., Ravagnani A.: Covering radius of matrix codes endowed with the rank metric. SIAM J. Discret. Math. 31(2), 927–944 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Chen, H., Cramer, R., Goldwasser, S., de Haan, R., Vaikuntanathan, V.: Secure computation from random error correcting codes. In: Advances in cryptology—EUROCRYPT 2007, volume 4515 of Lecture Notes in Computer Science, pp. 291–310 (2007)

  7. 7.

    Couvreur A., Márquez-Corbella I., Pellikaan R.: Cryptanalysis of McEliece cryptosystem based on algebraic geometry codes and their subcodes. IEEE Trans. Inf. Theory 63(8), 5404–5418 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    De Boer M.A.: Almost MDS codes. Des. Codes Cryptogr. 9(2), 143–155 (1996).

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    De la Cruz J., Gorla E., López H.H., Ravagnani A.: Weight distribution of rank-metric codes. Des. Codes Cryptogr. 86(1), 1–16 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25(3), 226–241 (1978).

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Dodunekova R., Dodunekov S.M., Kløve T.: Almost-MDS and near-MDS codes for error detection. IEEE Trans. Inf. Theory 43(1), 285–290 (1997).

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Ducoat, J.: Generalized rank weights: a duality statement. In: Topics in Finite Fields, volume 632 of Comtemporary Mathematics, pp. 114–123 (2015)

  13. 13.

    Feng G.-L., Rao T.R.N.: A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inf. Theory 40(4), 1003–1012 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Forney Jr. G.D.: Dimension/length profiles and trellis complexity of linear block codes. IEEE Trans. Inf. Theory 40(6), 1741–1752 (1994).

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Freij-Hollanti R., Gnilke O.W., Hollanti C., Karpuk D.A.: Private information retrieval from coded databases with colluding servers. SIAM J. Appl. Algebra Geom. 1(1), 647–664 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Inf. Trans. 21(1), 1–12 (1985).

    MathSciNet  MATH  Google Scholar 

  17. 17.

    El Gamal H., Hammons A.R.: On the design of algebraic space-time codes for MIMO block-fading channels. IEEE Trans. Inf. Theory 49(1), 151–163 (2003).

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Gopalan P., Huang C., Jenkins B., Yekhanin S.: Explicit maximally recoverable codes with locality. IEEE Trans. Inf. Theory 60(9), 5245–5256 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Greferath M., Honold T., Mc Fadden C., Wood J.A., Zumbrägel J.: Macwilliams’ extension theorem for bi-invariant weights over finite principal ideal rings. J. Comb. Theory Ser. A 125, 177–193 (2014).

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Hamming R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29(2), 147–160 (1950).

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Helleseth T., Kløve T., Levenshtein V.I., Ytrehus Ø.: Bounds on the minimum support weights. IEEE Trans. Inf. Theory 41(2), 432–440 (1995).

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Helleseth T., Kløve T., Mykkeltveit J.: The weight distribution of irreducible cyclic codes with block lengths \(n_l((q^l-1)/{N})\). Discret. Math. 18(2), 179–211 (1977).

    MATH  Article  Google Scholar 

  23. 23.

    Horlemann-Trautmann A.-L., Marshall K.: New criteria for MRD and Gabidulin codes and some rank-metric code constructions. Adv. Math. Commun. 11(3), 533 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Jurrius R., Pellikaan R.: On defining generalized rank weights. Adv. Math. Commun. 11(1), 225–235 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Jurrius R., Pellikaan R.: Defining the q-analogue of a matroid. Electron. J. Comb. 25, 321–330 (2018).

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Kötter, R.: A unified description of an error locating procedure for linear codes. In: Proceeding of the Algebraic and Combinatorial Coding Theory, pp. 113–117 (1992)

  27. 27.

    Kötter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54(8), 3579–3591 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Kurihara J., Matsumoto R., Uyematsu T.: Relative generalized rank weight of linear codes and its applications to network coding. IEEE Trans. Inf. Theory 61(7), 3912–3936 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Lam T.Y.: A general theory of Vandermonde matrices. Expos. Math. 4, 193–215 (1986).

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Lam T.Y., Leroy A.: Vandermonde and Wronskian matrices over division rings. J. Algebra 119(2), 308–336 (1988).

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Lu H.-F., Kumar P.V.: A unified construction of space-time codes with optimal rate-diversity tradeoff. IEEE Trans. Inf. Theory 51(5), 1709–1730 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Luo Y., Mitrpant C., Han Vinck A.J., Chen K.: Some new characters on the wire-tap channel of type II. IEEE Trans. Inf. Theory 51(3), 1222–1229 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    MacWilliams, F.J.: Combinatorial problems of elementary abelian groups. PhD thesis, Harvard (1962)

  34. 34.

    Mahmood R., Badr A., Khisti A.: Convolutional codes with maximum column sum rank for network streaming. IEEE Trans. Inf. Theory 62(6), 3039–3052 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Martínez-Peñas U.: On the similarities between generalized rank and Hamming weights and their applications to network coding. IEEE Trans. Inf. Theory 62(7), 4081–4095 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Martínez-Peñas, U.: Linearized multivariate skew polynomials and Hilbert 90 theorems with multivariate norms, In: Proceeding of the XVI EACA, Zaragoza-Encuentros de Álgebra Computacional y Aplicaciones, pp. 119–122 (2018)

  37. 37.

    Martínez-Peñas U.: Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring. J. Algebra 504, 587–612 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Martínez-Peñas, U.: Private information retrieval from locally repairable databases with colluding servers. (2019). Preprint: arXiv:1901.02938.

  39. 39.

    Martínez-Peñas, U., Kschischang, F.R.: Reliable and secure multishot network coding using linearized Reed–Solomon codes. In: Proceeding of the 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton), (2018). Extended version: arXiv:1805.03789

  40. 40.

    Martínez-Peñas, U., Kschischang, F.R.: Universal and dynamic locally repairable codes with maximal recoverability via sum-rank codes. In: Proceeding of the 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton), (2018). Extended version: arXiv:1809.11158.

  41. 41.

    Martínez-Peñas, U., Kschischang, F.R.: Evaluation and interpolation over multivariate skew polynomial rings. J. Algebra, pp. 1–28, (2019). In press. Available: arXiv:1710.09606

  42. 42.

    Martínez-Peñas U., Pellikaan R.: Rank error-correcting pairs. Des. Codes Cryptogr. 84(1–2), 261–281 (2017).

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Martínez-Peñas U., Matsumoto R.: Relative generalized matrix weights of matrix codes for universal security on wire-tap networks. IEEE Trans. Inf. Theory 64(4), 2529–2549 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Napp, D., Pinto, R., Rosenthal, J., Vettori, P.: MRD rank metric convolutional codes. In: Proceeding of the 2017 IEEE International Symposium on Information Theory, pp. 2766–2770 (2017)

  45. 45.

    Napp D., Pinto R., Sidorenko V.: Concatenation of convolutional codes and rank metric codes for multi-shot network coding. Des. Codes Cryptogr. 86(2), 303–318 (2018).

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Nóbrega, R.W., Uchôa-Filho, B.F.: Multishot codes for network coding: bounds and a multilevel construction. In: Proceeding of the 2009 IEEE International Symposium on Information Theory, pp. 428–432 (2009)

  47. 47.

    Nóbrega, R.W., Uchôa-Filho, B.F.: Multishot codes for network coding using rank-metric codes. In: Proceeding of the 2010 Third IEEE International Workshop on Wireless Network Coding, pp. 1–6 (2010)

  48. 48.

    Oggier, F.E., Sboui, A.: On the existence of generalized rank weights. In: Proceeding of the 2012 International Symposium on Information Theory and its Applications, pp. 406–410 (2012)

  49. 49.

    Ore O.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933).

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Overbeck R.: Structural attacks for public key cryptosystems based on Gabidulin codes. J. Cryptol. 21(2), 280–301 (2008).

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Ozarow, L.H., Wyner, A.D.: Wire-tap channel II. In: advances in cryptology: EUROCRYPT 84, volume 209 of Lecture Notes in Computer Science, pp. 33–50 (1985)

  52. 52.

    Pellikaan R.: On the existence of error-correcting pairs. J. Stat. Plan. Inference 51(2), 229–242 (1996).

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Ravagnani A.: Generalized weights: an anticode approach. J. Pure Appl. Algebra 220(5), 1946–1962 (2016).

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Roth R.M.: Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inf. Theory 37(2), 328–336 (1991).

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Silva D., Kschischang F.R.: Universal secure network coding via rank-metric codes. IEEE Trans. Inf. Theory 57(2), 1124–1135 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Singleton R.: Maximum distance q-nary codes. IEEE Trans. Inf. Theory 10(2), 116–118 (1964).

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Tsfasman M.A., Vlăduţ S.G.: Geometric approach to higher weights. IEEE Trans. Inf. Theory 41(6, part 1), 1564–1588 (1995).

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Wachter A., Sidorenko V.R., Bossert M., Zyablov V.V.: On (partial) unit memory codes based on Gabidulin codes. Probl. Inf. Trans. 47(2), 117–129 (2011).

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Wachter-Zeh A., Stinner M., Sidorenko V.: Convolutional codes in rank metric with application to random network coding. IEEE Trans. Inf. Theory 61(6), 3199–3213 (2015).

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inf. Theory 37(5), 1412–1418 (1991).

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

The author wishes to thank Frank R. Kschischang for valuable discussions on this manuscript. The author also whishes to thank the anonymous reviewers for the valuable comments on this work.

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A The skew metric and skew supports

A The skew metric and skew supports

In this Appendix, we briefly revisit the relation between the sum-rank metric (Definition 1) and the skew metric introduced in [37, Def. 9]. We extend such a relation to sum-rank supports and skew supports (which we introduce in this Appendix), and the corresponding support spaces. The exposition in this appendix follows the lines in [37].

Let \( \sigma : \mathbb {F} \longrightarrow \mathbb {F} \) be a field endomorphism and let \( \delta : \mathbb {F} \longrightarrow \mathbb {F} \) be a \( \sigma \)-derivation, that is, \( \delta \) is additive and \( \delta (ab) = \sigma (a) \delta (b) + \delta (a) b \), for all \( a,b \in \mathbb {F} \). Define the skew polynomial ring \( \mathbb {F}[x; \sigma , \delta ] \) as the vector space over \( \mathbb {F} \) with basis \( \{ x^i \mid i \in \mathbb {N} \} \) and with product given by the rules \( x^i x^j = x^{i + j} \), for \( i,j \in \mathbb {N} \), and

$$\begin{aligned} xa = \sigma (a) x + \delta (a), \end{aligned}$$
(5)

for \( a \in \mathbb {F} \). Define the degree of a non-zero skew polynomial \( F = \sum _{i \in \mathbb {N}} F_i x^i \in \mathbb {F}[x; \sigma , \delta ] \), denoted by \( \deg (F) \), as the maximum \( i \in \mathbb {N} \) such that \( F_i \ne 0 \). We also define \( \deg (0) = \infty \). Skew polynomial rings were introduced by Ore in [49] and the products given by (5) are the only products in \( \mathbb {F}[x; \sigma , \delta ] \) such that \( \deg (FG) = \deg (F) + \deg (G) \), for \( F,G \in \mathbb {F}[x; \sigma , \delta ] \). The extension to several variables was recently given in [41]. Conventional polynomial rings are recovered by setting \( \sigma = \mathrm{Id} \) and \( \delta = 0 \).

Since \( \mathbb {F}[x ; \sigma , \delta ] \) is a right Euclidean domain, we may define the evaluation of \( F \in \mathbb {F}[x; \sigma , \delta ] \) in \( a \in \mathbb {F} \) as the unique \( F(a) \in \mathbb {F} \) such that there exists \( G \in \mathbb {F}[x; \sigma , \delta ] \) with

$$\begin{aligned} F = G \cdot (x-a) + F(a). \end{aligned}$$

This concept of evaluation was introduced in [29, 30].

Given a subset \( \varOmega \subseteq \mathbb {F} \), we may define its associated ideal as \( I(\varOmega ) = \{ F \in \mathbb {F}[x;\sigma , \delta ] \mid F(a) = 0, \forall a \in \varOmega \} \). Observe that \( I(\varOmega ) \) is a left ideal in \( \mathbb {F}[x;\sigma , \delta ] \). Since \( \mathbb {F}[x;\sigma , \delta ] \) is a right Euclidean domain, there exists a unique monic skew polynomial \( F_\varOmega \in I(\varOmega ) \) of minimal degree among those in \( I(\varOmega ) \), which in turn generates \( I(\varOmega ) \) as left ideal. Such a skew polynomial is called the minimal skew polynomial of \( \varOmega \) [30].

Next, given a subset \( \varOmega \subseteq \mathbb {F} \), we define its P-closure as \( \overline{\varOmega } = Z(F_\varOmega ) \subseteq \mathbb {F} \) (the set of zeros of \( F_\varOmega \)), and we say that \( \varOmega \) is P-closed if \( \overline{\varOmega } = \varOmega \). A set \( \varOmega \subseteq \mathbb {F} \) is called P-independent if \( a \notin \overline{\varOmega \setminus \{ a \}} \), for all \( a \in \varOmega \). We say that \( \mathcal {B} \subseteq \varOmega \) is a P-basis of a P-closed set \( \varOmega \) if \( \mathcal {B} \) is P-independent and \( \varOmega = \overline{\mathcal {B}} \). We also say that \( \varOmega \) is a finitely generated P-closed set if it admits a finite P-basis, which is the case as long as \( \varOmega \ne \mathbb {F} \), or \( \varOmega = \mathbb {F} \) and \( \mathbb {F} \) is finite.

Given a finitely generated P-closed set \( \varOmega \subseteq \mathbb {F} \), any two of its P-bases are finite and have the same number of elements, which moreover coincides with \( \deg (F_\varOmega ) \). This motivates the definition rank of \( \varOmega \) as

$$\begin{aligned} \mathrm{Rk}(\varOmega ) = \deg (F_\varOmega ) < \infty . \end{aligned}$$

Fix a finitely generated P-closed set \( \varOmega \subseteq \mathbb {F} \) of rank n and fix one of its P-bases \( \mathcal {B} \). An important tool to define skew metrics is skew polynomial Lagrange interpolation. Let \( \mathbb {F}[x; \sigma , \delta ]_n \) be the n-dimensional vector space of skew polynomials of degree less than n. It follows from [29, Th. 8] that the evaluation map over the points in \( \mathcal {B} \),

$$\begin{aligned} E_\mathcal {B} : \mathbb {F}[x; \sigma , \delta ]_n \longrightarrow \mathbb {F}^\mathcal {B}, \end{aligned}$$

is a vector space isomorphism. Hence we may define skew weights [37, Def. 9] as follows.

Definition 15

(Skew weights [37]) Given \( F \in \mathbb {F}[x; \sigma , \delta ]_n \) and \( f = E_\mathcal {B}(F) \in \mathbb {F}^\mathcal {B} \), we define their skew weight over \( \varOmega \) as

$$\begin{aligned} \mathrm{wt}_\mathcal {B}(f) = \mathrm{wt}_\varOmega (F) = n - \mathrm{Rk}(Z_\varOmega (F)), \end{aligned}$$

where \( Z_\varOmega (F) = Z(F) \cap \varOmega = Z(\{ F, F_\varOmega \}) \) is the P-closed set of zeros of F in \( \varOmega \).

Skew weights are indeed weights [37, Prop. 10] and define a metric in \( \mathbb {F}^\mathcal {B} \), called the skew metric [37, Def. 11], by the usual formula: \( \mathrm{d}_\mathcal {B}(f, g) = \mathrm{wt}_\mathcal {B}(f - g) \), for \( f,g \in \mathbb {F}^\mathcal {B} \). To relate this metric with the sum-rank metric, we need the concept of conjugacy from [30]: We say that \( a, c \in \mathbb {F} \) are conjugates if there exists \( \beta \in \mathbb {F}^* \) such that

$$\begin{aligned} c = a^\beta {\mathop {=}\limits ^{def}} \sigma (\beta )\beta ^{-1} a + \delta (\beta )\beta ^{-1}. \end{aligned}$$

Putting together the results [29, Th. 23] and [30, Th. 4.5], we obtain the following characterization: A finite subset \( \mathcal {B} \subseteq \mathbb {F} \) with n elements is a P-basis of \( \varOmega = \overline{\mathcal {B}} \) if, and only if, \( n = n_1 + n_2 + \cdots + n_\ell \), for some \( \ell \), and there exists pair-wise non-conjugate elements \( a^{(1)}, a^{(2)}, \ldots , a^{(\ell )} \in \mathbb {F} \) and a set of linearly independent elements \( \{ \beta _1^{(i)}, \beta _2^{(i)}, \ldots , \beta _{n_i}^{(i)} \} \subseteq \mathbb {F} \), over the subfield \( K_i = K_{a^{(i)}} = \{ \beta \in \mathbb {F}^* \mid \left( a^{(i)} \right) ^{\beta } = a^{(i)} \} \cup \{ 0 \} \subseteq \mathbb {F} \), for each \( i = 1,2, \ldots , \ell \), such that

$$\begin{aligned} \mathcal {B} = \bigcup _{i = 1}^\ell \left\{ \left( a^{(i)} \right) ^{\beta _j^{(i)}} \mid j = 1,2, \ldots , n_i \right\} , \end{aligned}$$
(6)

where the union is disjoint. With this characterization at hand, we may give a vector space isomorphism connecting both metrics. The result follows from [37, Th. 2 & 3].

Theorem 6

[37] With notation as above, define the vector space isomorphism \( \phi _\mathcal {B} : \mathbb {F}^n \longrightarrow \mathbb {F}^\mathcal {B} \) by \( \phi _\mathcal {B} (\mathbf {c}^{(1)}, \)\( \mathbf {c}^{(2)}, \)\( \ldots , \)\( \mathbf {c}^{(\ell )}) = f \), where \( \mathbf {c}^{(i)} = (c_1^{(i)}, c_2^{(i)}, \ldots , c_{n_i}^{(i)}) \in \mathbb {F}^{n_i} \) and

$$\begin{aligned} f \left( \left( a^{(i)} \right) ^{\beta _j^{(i)}} \right) = c_j^{(i)} (\beta _j^{(i)})^{-1}, \end{aligned}$$
(7)

for \( j = 1,2, \ldots , n_i \) and \( i = 1,2,\ldots , \ell \). Then \( \phi _\mathcal {B} \) is an isometry: For \( \mathbf {c} \in \mathbb {F}^n \), it holds that

$$\begin{aligned} \mathrm{wt}_\mathcal {B}(\phi _\mathcal {B}(\mathbf {c})) = \mathrm{wt}_{SR}(\mathbf {c}), \end{aligned}$$

where \( \mathrm{wt}_{SR} \) is the sum-rank weight from Definition 1 with \( K_i = K_{a^{(i)}} \), for \( i = 1,2, \ldots , \ell \).

The representation (6) and the map given by (7) establish a dictionary between the sum-rank metric and the skew metric. This dictionary, however, depends on the conjugacy representatives \( a^{(1)}, a^{(2)}, \ldots , a^{(\ell )} \) and the P-basis \( \mathcal {B} \) of \( \varOmega \). The elements \( \beta _1^{(i)}, \beta _2^{(i)}, \)\( \ldots , \)\( \beta _{n_i}^{(i)} \) are determined up to scalar factor in \( K_i^* \) (thus uniquely as projective points in \( \mathbb {P}_{K_i}(\mathbb {F}) \)) by the conjugacy representatives and \( \mathcal {B} \), for \( i = 1,2, \ldots , \ell \). It is important to notice that in the case \( \sigma = \mathrm{Id} \) and \( \delta = 0 \), which corresponds to conventional polynomials and the Hamming metric, the dependency disappears since conjugacy classes only have one element and the only P-basis of \( \varOmega \) is \( \mathcal {B} = \varOmega \).

In particular, the concept of sum-rank support can be readily translated into the concept of skew support. First, define the lattice of skew supports in \( \varOmega \) as

$$\begin{aligned} \mathcal {P}_{Sk}(\varOmega ) = \{ \varPsi \subseteq \varOmega \mid \varPsi \text { is P-closed} \}. \end{aligned}$$

Thus skew supports will simply be P-closed subsets of \( \varOmega \), which form a lattice with intersections \( \varPsi _1 \cap \varPsi _2 \) and sums defined as \( \varPsi _1 + \varPsi _2 = \overline{\varPsi _1 \cup \varPsi _2} = Z(F_{\varPsi _1 \cup \varPsi _2}) \). The results [37, Prop. 43] and [37, Prop. 47] state that \( \mathcal {P}_{Sk}(\varOmega ) \) is a lattice isomorphic to \( \mathcal {P}(\mathbf {K}^\mathbf {n}) \), by mapping P-bases into lists of bases via (6), where \( K_i = K_{a^{(i)}} \), for \( i = 1,2, \ldots , \ell \). This mapping will now be used to define skew supports. As for vector and projective spaces, we implicitly associate the zero vector space with the empty P-closed set.

Definition 16

(Skew supports) With notation as above, let \( f \in \mathbb {F}^\mathcal {B} \) and define \( \mathbf {c} = (\mathbf {c}^{(1)}, \mathbf {c}^{(2)}, \ldots , \mathbf {c}^{(\ell )}) = \phi _\mathcal {B}^{-1}(f) \in \mathbb {F}^n \), where \( \mathbf {c}^{(i)} \in \mathbb {F}^{n_i} \), for \( i = 1,2, \ldots , \ell \). Next, let \( \gamma _h^{(i)} = \sum _{j=1}^{n_i} c_{h,j}^{(i)} \beta _j^{(i)} \in \mathbb {F} \), where \( (c_{h,1}^{(i)}, c_{h,2}^{(i)}, \ldots , c_{h,n_i}^{(i)}) \in K_i^{n_i} \) form the rows of \( M_{\mathcal {A}_i}(\mathbf {c}^{(i)}) \in K_i^{m_i \times n_i} \), for \( h = 1,2, \ldots , m_i \), and let \( \mathcal {G}_i \subseteq \mathbb {F}^* \) be a basis of the vector space generated by \( \gamma _1^{(i)}, \gamma _2^{(i)}, \)\( \ldots , \)\( \gamma _{m_i}^{(i)} \subseteq \mathbb {F} \) over \( K_i \), for \( i = 1,2, \ldots , \ell \). Define the P-independent set

$$\begin{aligned} \mathcal {B}_f = \bigcup _{i = 1}^\ell \left\{ \left( a^{(i)} \right) ^{\gamma } \mid \gamma \in \mathcal {G}_i \right\} . \end{aligned}$$

We define the skew support of \( f \in \mathbb {F}^\mathcal {B} \) as

$$\begin{aligned} \mathrm{Supp}_{Sk}(f) = \varOmega _f = \overline{\mathcal {B}_f} \in \mathcal {P}_{Sk}(\varOmega ). \end{aligned}$$

Finally, for a vector subspace \( \mathcal {F} \subseteq \mathbb {F}^\mathcal {B} \), we define its skew support as

$$\begin{aligned} \mathrm{Supp}_{Sk}(\mathcal {F}) = \sum _{f \in \mathcal {F}} \mathrm{Supp}_{Sk}(f) \in \mathcal {P}_{Sk}(\varOmega ), \end{aligned}$$

which allows to define the skew weight of \( \mathcal {F} \) as \( \mathrm{wt}_\mathcal {B}(\mathcal {F}) = \mathrm{Rk}(\mathrm{Supp}_{Sk}(\mathcal {F})) \).

As it was the case with the map in (7), the skew support \( \mathrm{Supp}_{Sk}(f) \in \mathcal {P}_{Sk}(\varOmega ) \) depends only on the conjugacy representatives and the choice of P-basis \( \mathcal {B} \) of \( \varOmega \). To see this, note that the vector space generated by the rows of \( M_{\mathcal {A}_i}(\mathbf {c}^{(i)}) \) does not depend on \( \mathcal {A}_i \), and secondly, the P-basis corresponding to different bases of the subspace generated by \( \gamma _1^{(i)}, \gamma _2^{(i)}, \ldots , \gamma _{m_i}^{(i)} \in \mathbb {F} \) over \( K_i \) generate the same P-closed set \( \varOmega _f \) by [37, Cor. 27].

Using the same arguments, we may prove the following properties:

Proposition 11

The following properties hold.

  1. (1)

    For \( f \in \mathbb {F}^\mathcal {B} \) and \( a \in \mathbb {F}^* \), it holds that \( \mathrm{Supp}_{Sk}(af) = \mathrm{Supp}_{Sk}(\langle f \rangle ) = \mathrm{Supp}_{Sk}(f) \) and

    $$\begin{aligned} \mathrm{Rk}(\mathrm{Supp}_{Sk}(f)) = \mathrm{wt}_\mathcal {B}(f). \end{aligned}$$
  2. (2)

    \( \phi _\mathcal {B}^{-1}(f) \) and \( \phi _\mathcal {B}^{-1}(g) \) have the same sum-rank support if, and only if, f and g have the same skew support, for \( f,g \in \mathbb {F}^\mathcal {B} \). The same holds for subspaces of \( \mathbb {F}^\mathcal {B} \).

  3. (3)

    If \( \mathcal {F} \subseteq \mathbb {F}^\mathcal {B} \) and \( \mathcal {D} = \phi _\mathcal {B}^{-1}(\mathcal {F}) \subseteq \mathbb {F}^n \) are subspaces, then

    $$\begin{aligned} \mathrm{wt}_\mathcal {B}(\mathcal {F}) = \mathrm{Rk}(\mathrm{Supp}_{Sk}(\mathcal {F})) = \mathrm{Rk}(\mathrm{Supp}(\mathcal {D})) = \mathrm{wt}_{SR}(\mathcal {D}) . \end{aligned}$$

The concept of skew support space may also be considered. It may be introduced as a lattice of subspaces of \( \mathbb {F}^\mathcal {B} \).

Definition 17

(Skew support spaces) Given a P-closed subset \( \varPsi \subseteq \varOmega \) (i.e. \( \varPsi \in \mathcal {P}_{Sk}(\varOmega ) \)), we define the skew support space associated to \( \varPsi \) over \( \mathcal {B} \) as

$$\begin{aligned} \mathcal {W}_\varPsi = \{ f \in \mathbb {F}^\mathcal {B} \mid \mathrm{Supp}_{Sk}(f) \subseteq \varPsi \} \subseteq \mathbb {F}^\mathcal {B} . \end{aligned}$$

We may add to Theorem 1 the following characterizations. They follow from the results in this Appendix, except for the arithmetic characterizations in Items 3 and 4. These follow by combining Item 6 in Theorem 1 and the recent result [38, Th. 2], which gives the connection between coordinate-wise matrix products as in Theorem 1 and products of skew polynomials given by (5).

Proposition 12

The following are equivalent:

  1. (1)

    \( \mathcal {W} \) is a skew support space, that is, there exists \( \varPsi \in \mathcal {P}_{Sk}(\varOmega ) \) such that \( \mathcal {W} = \mathcal {W}_\varPsi \).

  2. (2)

    \( \mathcal {V} = \phi _\mathcal {B}^{-1}(\mathcal {W}) \subseteq \mathbb {F}^n \) is a sum-rank support space.

  3. (3)

    \( \mathcal {W} \) is a left ideal of \( \mathbb {F}^\mathcal {B} \) for the product in \( \mathbb {F}^\mathcal {B} \) given by \( f g \in \mathbb {F}^\mathcal {B} \), where

    $$\begin{aligned} (fg)(a) = (FG)(a), \end{aligned}$$
    (8)

    for \( a \in \mathcal {B} \), \( f,g \in \mathbb {F}^\mathcal {B} \) and \( F,G \in \mathbb {F}[x; \sigma , \delta ]_n \) such that \( f = E_\mathcal {B}(F) \) and \( g = E_\mathcal {B}(G) \).

  4. (4)

    There exists a P-closed subset \( \varPhi \subseteq \varOmega \) such that \( \mathcal {W} = E_\mathcal {B}(I(\varPhi )) \).

In particular, by Item 2, skew support spaces are also vector subspaces of \( \mathbb {F}^\mathcal {B} \). Notice also that, in general, \( (fg)(a) \ne f(a) g(a) \) in Item 3 (see [30, Th. 2.7]).

In conclusion, in this Appendix we have introduced skew supports and support spaces, and we have given the precise connections with sum-rank supports and support spaces. Except for the Hamming-metric case, the dictionary between both types of concepts depends on the choice of conjugacy representatives and P-basis of the ambient P-closed set via (6). With this dictionary, all of the remaining results and definitions in this paper can be translated to skew supports and support spaces. We leave however as open problem defining skew supports and support spaces independently of a set of conjugacy representatives and a P-basis.

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Martínez-Peñas, U. Theory of supports for linear codes endowed with the sum-rank metric. Des. Codes Cryptogr. 87, 2295–2320 (2019). https://doi.org/10.1007/s10623-019-00619-8

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Keywords

  • Generalized sum-rank weights
  • Hamming metric
  • MSRD codes
  • Multishot matrix-multiplicative channel
  • Rank metric
  • Sum-rank metric
  • Sum-rank support
  • Wire-tap channel

Mathematics Subject Classification

  • 94A60
  • 94B05
  • 94C99