329 research outputs found
Ramadurai: Higher Weights of Codes from Projective Planes and Biplanes Produced by The Berkeley Electronic Press,
No full textAbstract We study the higher weights of codes formed from planes and biplanes. We relate the higher weights of the Hull and the code of a plane and biplane. We determine all higher weight enumerators of planes and biplanes of order less or equal to 4. HIGHER WEIGHTS OF CODES FROM PROJECTIVE PLANES AND BIPLANES Steven T. DOUGHERTY and Reshma RAMADURAI Abstract. We study the higher weights of codes formed from planes and biplanes. We relate the higher weights of the Hull and the code of a plane and biplane. We determine all higher weight enumerators of planes and biplanes of order less or equal to 4
Composite Constructions of Self-Dual Codes from Group Rings and New Extremal Self-Dual Binary Codes of Length 68
No full textThis is a pre-copy-editing, author-produced PDF of an article accepted for publication in Advances in Mathematics of Communications following peer review. The definitive publisher-authenticated version Dougherty, S, T., Gildea, J., Korban, A. & Kaya, A. (2019 - forthcoming). Composite Constructions of Self-Dual Codes from Group Rings and New Extremal Self-Dual Binary Codes of Length 68. Advances in Mathematics of Communications, is available online at: 10.3934/amc.2020037.We describe eight composite constructions from group rings where the orders of
the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over
F4. These codes have binary images with parameters [32, 16, 8] or [32, 16, 6]. These
are lifted to codes over F4 + uF4, to obtain codes with Gray images extremal self-dual
binary codes of length 64. Finally, we use a building-up method over F2 + uF2 to
obtain new extremal binary self-dual codes of length 68. We construct 11 new codes
via the building-up method and 2 new codes by considering possible neighbors
Algebraic coding theory over finite commutative rings
No full textThis book provides a self-contained introduction to algebraic coding theory over finite Frobenius rings. It is the first to offer a comprehensive account on the subject. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is generally the binary field. Since its inception, it has grown as a branch of mathematics, and has since been expanded to consider any finite field, and later also Frobenius rings, as its alphabet. This book presents a broad view of the subject as a branch of pure mathematics and relates major results to other fields, including combinatorics, number theory and ring theory. Suitable for graduate students, the book will be of interest to anyone working in the field of coding theory, as well as algebraists and number theorists looking to apply coding theory to their own work
G-codes over Formal Power Series Rings and Finite Chain Rings
No full textIn this work, we define -codes over the infinite ring as ideals in the group ring . We show that the dual of a -code is again a -code in this setting. We study the projections and lifts of -codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing -adic codes over to -adic -codes over the same ring. We also study -codes over principal ideal rings
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