Ranks and kernels of codes from generalized Hadamard matrices

The ranks and kernels of generalized Hadamard matrices are studied. It is proved that any generalized Hadamard matrix H(q, λ) over Fq , q > 3, or q = 3 and gcd(3, λ) ≠ 1, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed

Linearity and classification of ZpZp^2-linear generalized Hadamard codes

The ZpZp2 -additive codes are subgroups of Zα1 p × Zα2 p2 , and can be seen as linear codes over Zp when α2 = 0, Zp2 -additive codes when α1 = 0, or Z2Z4-additive codes when p = 2. A ZpZp2 -linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2 -additive code. Recursive constructions of ZpZp2 -additive GH codes of type (α1, α2;t1,t2) with t1,t2 ≥ 1 are known. In this paper, we generalize some known results for ZpZp2 -linear GH codes with p = 2 to any p ≥ 3 prime when α1 = 0, and then we compare them with the ones obtained when α1 = 0. First, we show for which types the corresponding ZpZp2 -linear GH codes are nonlinear over Zp. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike Z4-linear Hadamard codes, the Zp2 -linear GH codes are not included in the family of ZpZp2 - linear GH codes with α1 = 0 when p ≥ 3 prime. Indeed, there are some families with infinite nonlinear ZpZp2 -linearGH codes, where the codes are not equivalent to any Zps - linear GH code with s ≥ 2

On the constructions of ZpZp2-linear generalized Hadamard codes

Altres ajuts: acord transformatiu CRUE-CSICThe ZZ-additive codes are subgroups of Z ×Z , and can be seen as linear codes over Z when α=0, Z-additive codes when α=0, or ZZ-additive codes when p=2. A ZZ-linear generalized Hadamard (GH) code is a GH code over Z which is the Gray map image of a ZZ-additive code. In this paper, we generalize some known results for ZZ-linear GH codes with p=2 to any p≥3 prime when α≠0. First, we give a recursive construction of ZZ-additive GH codes of type (α,α;t,t) with t,t≥1. We also present many different recursive constructions of ZZ-additive GH codes having the same type, and show that we obtain permutation equivalent codes after applying the Gray map. Finally, according to some computational results, we see that, unlike Z-linear GH codes, when p≥3 prime, the Z-linear GH codes are not included in the family of ZZ-linear GH codes with α≠0. Indeed, we observe that the constructed codes are not equivalent to the Z-linear GH codes for any s≥2

Ranks and kernels of codes from generalized Hadamard matrices

The ranks and kernels of generalized Hadamard matrices are studied. It is proved that any generalized Hadamard matrix H(q, λ) over Fq , q > 3, or q = 3 and gcd(3, λ) ≠ 1, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed