A new family of linear maximum rank distance codes

Sheekey, John

doi:10.3934/amc.2016019

Cited by 208 publications

(345 citation statements)

References 40 publications

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“…Using the strategy of this algorithm we will describe the automorphism groups of Gabidulin-like codes in Section 4. In the special case of classical Gabidulin codes of full length m = ℓ these groups have already been determined in [10] and [14].…”

Section: Introductionmentioning

confidence: 99%

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Automorphism groups of Gabidulin-like codes

Liebhold

Nebe

2016

Arch. Math.

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Abstract. Let K/k be a cyclic Galois extension of degree ℓ and θ a generator of Gal(K/k). For any v = (v 1 , . . . , v m ) ∈ K m such that v is linearly independent over k, and any 1 ≤ d < m the Gabidulin-like code C(v, θ, d) ≤ k ℓ×m is a maximum rank distance code of dimension ℓd over k. This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulinlike codes and determine their rank-metric automorphism group.

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Section: Introductionmentioning

confidence: 99%

“…These examples have the property that they are linear over the larger field. Until recently, [14], all known families of MRD codes arose from linear codes over some extension field K, so called lifted codes:…”

Section: Introductionmentioning

confidence: 99%

Automorphism groups of Gabidulin-like codes

Liebhold

Nebe

2016

Arch. Math.

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“…In , the following sets were introduced.

{scriptH}_{k, σ} false(η, ρ false) : = false{f_{0} X + f_{1} X^{σ} + \dots + f_{k - 1} X^{σ^{k - 1}} + η f_{0}^{ρ} X^{σ^{k}} : f_{i} \in L false} \subset {normalEnd}_{K} false(L false)

Taking

L = {double-struckF}_{q^{n}}

X^{σ} = X^{q}

η = 0

, returns the Gabidulin codes

G_{k, σ_{q}}

, while taking

L = {double-struckF}_{q^{n}}

X^{σ} = X^{q^{s}}

η = 0

returns the generalised Gabidulin codes of .…”

Section: Semifields and Maximum Rank Distance Codesmentioning

confidence: 99%

“…In , it was shown that for

x^{σ} = x^{q}

, and

f_{0}^{ρ} = f_{0}^{σ^{h}}

{scriptH}_{k} false(η, ρ false)

is an MRD‐code whenever

N false(η false) \neq {(- 1)}^{n k}

. In , this was extended by allowing

σ

to be any

F_{q}

‐automorphism of

F_{q^{n}}

, while in this was extended by allowing

ρ

to be any automorphism of

F_{q^{n}}

, not necessarily fixing

F_{q}

.…”

Section: Semifields and Maximum Rank Distance Codesmentioning

confidence: 99%

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New semifields and new MRD codes from skew polynomial rings

Sheekey

2019

J. London Math. Soc.

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In this article, we construct a new family of semifields, containing and extending two well‐known families, namely Albert's generalised twisted fields and Petit's cyclic semifields (also known as Johnson–Jha semifields). The construction also gives examples of semifields with parameters for which no examples were previously known. In the case of semifields two dimensions over a nucleus and four‐dimensional over their centre, the construction gives all possible examples. Furthermore we embed these semifields in a new family of maximum rank‐distance codes, encompassing most known current constructions, including the (twisted) Delsarte–Gabidulin codes, and containing new examples for most parameters.

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Constructing MRD codes by switching

Shi,

Krotov,

Özbudak

2024

J of Combinatorial Designs

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Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on ‐by‐ matrices over a finite field . They are diameter perfect and have the cardinality if . We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in if the other parameters (, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.

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A new family of linear maximum rank distance codes

Cited by 208 publications

References 40 publications

Automorphism groups of Gabidulin-like codes

Automorphism groups of Gabidulin-like codes

New semifields and new MRD codes from skew polynomial rings

Constructing MRD codes by switching

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