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ADDITIVE SELF-DUAL CODES OVER FIELDS OF EVEN ORDER
Dougherty, Steven T.,Kim, Jon-Lark,Lee, Nari Korean Mathematical Society 2018 대한수학회보 Vol.55 No.2
We examine various dualities over the fields of even orders, giving new dualities for additive codes. We relate the MacWilliams relations and the duals of ${\mathbb{F}}_{2^{2s}}$ codes for these various dualities. We study self-dual codes with respect to these dualities and prove that any subgroup of order $2^s$ of the additive group is a self-dual code with respect to some duality.
Additive self-dual codes over fields of even order
Steven T. Dougherty,김종락,이나리 대한수학회 2018 대한수학회보 Vol.55 No.2
We examine various dualities over the fields of even orders, giving new dualities for additive codes. We relate the MacWilliams relations and the duals of $\FF_{2^{2s}}$ codes for these various dualities. We study self-dual codes with respect to these dualities and prove that any subgroup of order $2^s$ of the additive group is a self-dual code with respect to some duality.
OPTIMAL LINEAR CODES OVER ℤ<sub>m</sub>
Dougherty, Steven T.,Gulliver, T. Aaron,Park, Young-Ho,Wong, John N.C. Korean Mathematical Society 2007 대한수학회지 Vol.44 No.5
We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring ${\mathbb{Z}}_m$. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over ${\mathbb{Z}}_8$ and ${\mathbb{Z}}_9$ of lengths up to 6. We determine the minimum distances of optimal linear codes over ${\mathbb{Z}}_4$ for lengths up to 7. Some examples of optimal codes are given.
HIGHER WEIGHTS AND GENERALIZED MDS CODES
Dougherty, Steven T.,Han, Sung-Hyu Korean Mathematical Society 2010 대한수학회지 Vol.47 No.6
We study codes meeting a generalized version of the Singleton bound for higher weights. We show that some of the higher weight enumerators of these codes are uniquely determined. We give the higher weight enumerators for MDS codes, the Simplex codes, the Hamming codes, the first order Reed-Muller codes and their dual codes. For the putative [72, 36, 16] code we find the i-th higher weight enumerators for i = 12 to 36. Additionally, we give a version of the generalized Singleton bound for non-linear codes.
HIGHER WEIGHTS AND GENERALIZED MDS CODES
Steven T. Dougherty,한성휴 대한수학회 2010 대한수학회지 Vol.47 No.6
We study codes meeting a generalized version of the Singleton bound for higher weights. We show that some of the higher weight enumerators of these codes are uniquely determined. We give the higher weight enumerators for MDS codes, the Simplex codes, the Hamming codes, the first order Reed-Muller codes and their dual codes. For the putative [72, 36, 16] code we find the i-th higher weight enumerators for i = 12 to 36. Additionally, we give a version of the generalized Singleton bound for non-linear codes.