This PhD thesis is focused on generalized Hadamard (GH) codes obtained from additive codes over Z_{p^s} and some mixed alphabets. GH codes are (n,pn,n(p-1)/p) codes, which are optimal with respect to Plotkin bound. The classification of nonlinear GH codes is still an open problem, which is far from being solved. The number of nonequivalent GH codes is only known for small sizes. In this PhD thesis, the main goal is to classify GH codes of a given size. Even though the full classification is far from being established, we present new results in this direction. The ZpZp^2. . . Zp^s-additive codes are subgroups of Z_p^{alpha_1} X Z_{p^2}^{alpha_2} X. . . X Z_{p^s}^{alpha_s}. A ZpZp^2. . . Zp^s-linear GH code is a generalized Hadamard code over Zp which is the Gray map image of a ZpZp^2. . . Zp^s-additive code. For p=2, we write Hadamard instead of GH in the definitions. We generalize some known results for Z_{2^s}-linear and Z2Z4-linear Hadamard codes to Z_{p^s}-linear GH codes with p>=3 prime and ZpZp^2. . . Zp^s-linear GH codes with p>=3 prime when s=2 and p=2 when s=3. We describe recursive constructions for some families of these codes of type (alpha_1, . . . , alpha_s;t_1,. . . , t_s). It is shown for which types the corresponding ZpZp^2. . . Zp^s-linear GH codes of length p^t are nonlinear. For these codes, the rank and dimension of the kernel, which allow us to give a partial classification of these codes, are computed. In some cases, a complete classification can be provided, by giving the exact amount of nonequivalent such codes for a given length. The equivalence relations between several infinite families of these codes are studied. We give some families with infinite nonlinear ZpZp^2-linear GH codes which are not equivalent to any Z_{p^s}-linear GH code with s>=2. We also prove the existence of several families of infinite such nonlinear Z2Z4Z8-linear Hadamard codes, which are not equivalent to any other constructed Z2Z4Z8-linear Hadamard code, nor to any Z2Z4-linear Hadamard code, nor to any previously constructed Z_{2^s}-linear Hadamard code with s>=2, with the same length 2^t.
Classification of Generalized Hadamard Codes Obtained From Additive Codes Over Z/p^s and Some Mixed Alphabets
Bhunia, D. K. (Author). 20 Mar 2024
Student thesis: Doctoral thesis
Student thesis: Doctoral thesis