Group rings, \(G\)-codes and constructions of self-dual and formally self-dual codes
From MaRDI portal
Publication:1650820
DOI10.1007/s10623-017-0440-7zbMath1411.94098arXiv1604.07863OpenAlexW2767540775MaRDI QIDQ1650820
Joseph Gildea, Rhian Taylor, Steven T. Dougherty, Alexander Tylyshchak
Publication date: 13 July 2018
Published in: Designs, Codes and Cryptography (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/1604.07863
Algebraic coding theory; cryptography (number-theoretic aspects) (11T71) Linear codes (general theory) (94B05) Group rings (16S34) Cyclic codes (94B15)
Related Items (33)
\(G\)-codes, self-dual \(G\)-codes and reversible \(G\)-codes over the ring \(\mathscr{B}_{j,k} \) ⋮ Reversible \(G^k\)-codes with applications to DNA codes ⋮ Construction and enumeration of left dihedral codes satisfying certain duality properties ⋮ Group LCD and group reversible LCD codes ⋮ Additive \(G\)-codes over \(\mathbb{F}_q\) and their dualities ⋮ Quaternary group ring codes: ranks, kernels and self-dual codes ⋮ Quadruple bordered constructions of self-dual codes from group rings ⋮ Unnamed Item ⋮ Reversible \(G\)-codes over the ring \(\mathcal{F}_{j, k}\) with applications to DNA codes ⋮ On the algebraic structure of quasi-group codes ⋮ Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes ⋮ New quantum codes derived from group rings ⋮ New type I binary \([72, 36, 12\) self-dual codes from composite matrices and \(R_1\) lifts] ⋮ Additive skew \(G\)-codes over finite fields ⋮ Codes from the Skew Ring 𝑀₂(𝔽₂)⋊_{𝜑}𝔾 ⋮ New self-dual codes from \(2 \times 2\) block circulant matrices, group rings and neighbours of neighbours ⋮ Self-dual codes using bisymmetric matrices and group rings ⋮ New binary quantum codes from group rings and skew group rings ⋮ Group matrix ring codes and constructions of self-dual codes ⋮ Double bordered constructions of self-dual codes from group rings over Frobenius rings ⋮ Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68 ⋮ \(2^n\) bordered constructions of self-dual codes from group rings ⋮ Constructing self-dual codes from group rings and reverse circulant matrices ⋮ Composite matrices from group rings, composite \(G\)-codes and constructions of self-dual codes ⋮ Self-dual codes over a family of local rings ⋮ Composite G-codes over formal power series rings and finite chain rings ⋮ Extending an established isomorphism between group rings and a subring of the n × n matrices ⋮ An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes. I ⋮ New self-dual codes of length 68 from a \(2 \times 2\) block matrix construction and group rings ⋮ Hermitian duality of left dihedral codes over finite fields ⋮ A modified bordered construction for self-dual codes from group rings ⋮ Maximal entanglement-assisted quantum error correction codes from the skew group ring \(\mathbb{F}_4\rtimes_\varphi G\) by a heuristic search scheme ⋮ Skew G-codes
Uses Software
Cites Work
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Quasi-cyclic codes as cyclic codes over a family of local rings
- An extremal \([72,36,16\) binary code has no automorphism group containing \(Z_2 \times Z_4\), \(Q_8\), or \(Z_{10}\)]
- Constructions of self-dual codes and formally self-dual codes over rings
- Codes over \(R_k\), Gray maps and their binary images
- A group ring construction of the \([48,24,12\) Type II linear block code]
- Cyclic codes over \(R_{k}\)
- The extended Golay codes considered as ideals
- The automorphism group of a self-dual \([72,36,16\) code is not an elementary abelian group of order 8]
- The Automorphism Group of a Binary Self-Dual Doubly Even$[72,36,16$Code is Solvable]
- A Group Ring Construction of the Extended Binary Golay Code
- A Putative Doubly Even [72,36,16 Code Does Not Have an Automorphism of Order 9]
- Algebraic Coding Theory Over Finite Commutative Rings
- On the Automorphism Group of a Binary Self-Dual Doubly Even $[{72, 36, 16}$ Code]
- Binary Codes Which Are Ideals in the Group Algebra of an Abelian Group
- On the automorphisms of order 2 with fixed points for the extremal self-dual codes of length 24\(m\)
This page was built for publication: Group rings, \(G\)-codes and constructions of self-dual and formally self-dual codes