Abstract In this paper, we propose a new erasure decoding algorithm for convolutional codes using the generator matrix. This implies that our decoding method also applies to catastrophic convolutional codes in opposite to the classic approach using the parity-check matrix. We compare the performance of both decoding algorithms. Moreover, we enlarge the family of optimal convolutional codes (complete-MDP) based on the generator matrix.

The Zps-additive codes of lengthnare subgroups of Znps, withpprime ands≥ 1. They can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s. In this paper, we show two methods for computing a parity-check matrix of a Zps-additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for linear codes over finite rings in general. Complementing this comparison, we also show a time complexity analysis of the algorithms. The rings Zpsbelong to a more general class of rings: finite chain rings. Along the paper, we observe that the same results can be applied to any linear code over a finite commutative chain ring.

The$\mathbb{Z}_{p^{s}}-\mathbf{additive}$codes of length$n$are subgroups of$\mathbb{Z}_{p^{s}}^{n}$, with$p$prime and$s\geq 1$. They can be seen as a generalization of linear codes over$\mathbb{Z}_{2},\ \mathbb{Z}_{4}$, or more general over$\mathbb{Z}_{2^{s}}$. In this paper, we show two methods for computing a parity-check matrix of a$\mathbb{Z}_{p^{s}} -\mathbf{additive}$code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general.

Abstract Linear codes of length n over $$\mathbb {Z}_{p^s}$$ Z p s , p prime, called $$\mathbb {Z}_{p^s}$$ Z p s -additive codes, can be seen as subgroups of $$\mathbb {Z}_{p^s}^n$$ Z p s n . A $$\mathbb {Z}_{p^s}$$ Z p s -linear generalized Hadamard (GH) code is a GH code over $$\mathbb {Z}_p$$ Z p which is the image of a $$\mathbb {Z}_{p^s}$$ Z p s -additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for $$p\ge 3$$ p ≥ 3 prime, we establish that some $$\mathbb {Z}_{p^s}$$ Z p s -linear GH codes of length $$p^t$$ p t having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to $$t=10$$ t = 10 if $$p=3$$ p = 3 or $$t=8$$ t = 8 if $$p=5$$ p = 5 , this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel.

$\mathbb {Z}_{2^{s}}$-additive codes are subgroups of$\mathbb {Z}^{n}_{2^{s}}$, and can be seen as a generalization of linear codes over$\mathbb {Z}_{2}$and$\mathbb {Z}_{4}$. A$\mathbb {Z}_{2^{s}}$-linear code is a binary code (not necessarily linear) which is the Gray map image of a$\mathbb {Z}_{2^{s}}$-additive code. We consider$\mathbb {Z}_{2^{s}}$-additive simplex codes of type$\alpha $and$\beta $, which are a generalization over$\mathbb {Z}_{2^{s}}$of the binary simplex codes. These codes are related to the$\mathbb {Z}_{2^{s}}$-additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding$\mathbb {Z}_{2^{s}}$-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly,$\mathbb {Z}_{2^{s}}$-additive MacDonald codes are defined for$s>2$, and equivalent results are obtained.

Z2$s$-additive codes are subgroups of Zn2s, and can be seen as a generalization of linear codes over Z2and Z4. A Z2$s$-linear code is a binary code (not necessarily linear) which is the Gray map image of a Z2$s$-additive code. We consider Z2s- additive simplex codes of type a and β, which are a generalization over$Z$2$s$of the binary simplex codes. These codes are related to the Z2$s$-additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding$Z$2s-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, Z2$s$-additive MacDonald codes are defined for$s$> 2, and equivalent results are obtained.

The Z2(s)-additive codes are subgroups of Zn2(s), and can be seen as a generalization of linear codes over Z2and Z4. A Z2(s)-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2(s)-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z2(s)-linear Hadamard codes of length 2tonly provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established.