Extending two families of maximum rank distance codes

Abstract

In this paper we provide a large family of rank-metric codes, which contains properly the codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are F q 2t -linear of dimension 2 in the space of linearized polynomials over F q 2t , where t is any integer greater than 2, and we prove that they are maximum rank distance codes. For t ≥ 5, we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new.

Linear maximum rank distance codes of exceptional type

“…x q si : i / ∈ {0, 1, t − 1, t + 1, 2t − 1}, h1(x) = x q s − x q s(t−1) , h2(x) = δ q t +1 x q s − x q s(t+1) , h3(x) = δ 1−q 2t−1 x q s − x q s(2t−1) q odd, N q 2t (q t (δ) = −1, gcd(s, n) = 1 [7,24,25,31,40] 6 2…”

On a family of linear MRD codes with parameters $[8\times8,16,7]_q$

“…(2) C G(r) := x, x q r F q 8 with r ∈ {1, 3, 5, 7}, generalized Gabidulin codes (see [2]); (3) C T (ǫ,1) := x, ǫx q r +x q 8−r F q 8 with r ∈ {1, 3, 5, 7} and N q 8 /q (ǫ) / ∈ {0, 1}, generalized twisted Gabidulin codes (see [22,12]); (4) C N SZ(h,r) := x, ψ h,r (x) F q 8 , where ψ h,r (x) = x q r + x q 3r + h q r +1 x q 5r + h 1−q 7r x q 7r with r ∈ {1, 3, 5, 7} and h q 4 +1 = −1 (see [15], which generalizes [11] and [10]). We show that C δ,s is not equivalent to any code of type (2), ( 3) and ( 4), nor to their adjoint codes.…”

“…Lemma 4.1. (see the proof of [15,Theorem 4.6]) Let r ∈ {1, 3, 5, 7} and h ∈ F q 8 with h q 4 +1 = −1. Then there exists h ∈ F q 8 with hq 4 +1 = −1 such that C N SZ(h,r) is equivalent to C N SZ( h,1) .…”

“…Actually, Theorem 4.6 in [15] is stated for n = 2t with t ≥ 5. Yet, the arguments used in its proof to provide an explicit equivalence still hold in the case t = 4, hence proving the equivalence between C N SZ(h,r) and C N SZ( h,1) for some h ∈ F q 8 with hq 4 +1 = −1.…”

“…Most of the known families of F q -linear MRD codes are 2-dimensional F q n -linear subspaces of L n,q ; see e.g. [4,14,15] and the references therein. More precisely, up to a suitable equivalence, they are rank metric codes with parameters [n × n, 2n, d] q of the shape (1)…”