17 research outputs found
Algebraic lattice constellations: bounds on performance
In this work, we give a bound on performance of any full-diversity lattice constellation constructed from algebraic number fields. We show that most of the already available constructions are almost optimal in the sense that any further improvement of the minimum product distance would lead to a negligible coding gain. Furthermore, we discuss constructions, minimum product distance, and bounds for full-diversity complex rotated Z[i]/sup n/-lattices for any dimension n, which avoid the need of component interleaving
Construction and classification of Zâs-linear Hadamard codes
The Zâs-additive and ZâZâ-additive codes are subgroups of Zâs^n and Zâ^α Ă Zâ^ÎČ, respectively. Both families can be seen as generalizations of linear codes over Zâ and Zâ. A Zâs-linear (resp. ZâZâ-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Zâs-additive (resp. ZâZâ-additive) code. It is known that there are exactly â(tâ1)/2â and ât/2â nonequivalent ZâZâ-linear Hadamard codes of length 2á”, with α=0 and αâ 0, respectively, for all tâ„3. In this paper, new Zâs-linear Hadamard codes are constructed for s>2, which are not equivalent to any ZâZâ-linear Hadamard code. Moreover, for each s>2, it is claimed that the new constructed nonlinear Zâ-linear Hadamard codes of length 2á” are pairwise nonequivalent
On the automorphism groups of the ZâZâ-linear Hadamard codes and their classification
PublicaciĂł amb motiu del 4th International Castle Meeting (Pamela Castle, Portugal). Sep. 15-18 2014It is known that there are exactly â(tâ1)/2â and ât/2â nonequivalent ZâZâ-linear Hadamard codes of length 2á” , with αâ=â0 and αâ 0, respectively, for all tââ„â3. In this paper, it is shown that each ZâZâ-linear Hadamard code with αâ=â0 is equivalent to a ZâZâ-linear Hadamard code with αââ â0, so there are only ât/2â nonequivalent ZâZâ-linear Hadamard codes of length 2á”. Moreover, the orders of the permutation automorphism groups of the ZâZâ-linear Hadamard codes are given
On the Kernel of -Linear Hadamard Codes
The -additive codes are subgroups of ,
and can be seen as a generalization of linear codes over and
. A -linear Hadamard code is a binary Hadamard
code which is the Gray map image of a -additive code. It is
known that the dimension of the kernel can be used to give a complete
classification of the -linear Hadamard codes. In this paper, the
kernel of -linear Hadamard codes and its dimension are
established for . Moreover, we prove that this invariant only provides a
complete classification for some values of and . The exact amount of
nonequivalent such codes are given up to for any , by using
also the rank and, in some cases, further computations
Linearity and classification of ZpZp^2-linear generalized Hadamard codes
The ZpZp2 -additive codes are subgroups of Zα1 p à 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 equivalence of Z ps -linear generalized Hadamard codes
Altres ajuts: acords transformatius de la UABLinear codes of length n over Zps , p prime, called Zps -additive codes, can be seen as subgroups of Zn ps . A Zps -linear generalized Hadamard (GH) code is a GH code over Zp which is the image of a Zps -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 â„ 3 prime, we establish that some Zps -linear GH codes of length pt 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 if p = 3 or t = 8 if p = 5, this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel
On the linearity and classification of Z_p^s-linear generalized Hadamard codes
Acord transformatiu CRUE-CSICZp^s-additive codes of length n are subgroups of (Zp^s)^n , and can be seen as a generalization of linear codes over Z2, Z4 , or Z2^s in general. A Zp^s-linear generalized Hadamard (GH) code is a GH code over Zp which is the image of a Zp^s-additive code by a generalized Gray map. In this paper, we generalize some known results for Zp^s-linear GH codes with p = 2 to any odd prime p. First, we show some results related to the generalized Carlet's Gray map. Then, by using an iterative construction of Zp^s -additive GH codes of type (n; t 1 , . . . , t s ), we show for which types the corresponding Zp^s-linear GH codes of length p^t are nonlinear over Zp .For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for p â„ 3 are different from the case with p = 2. Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s, t, and any p â„ 2; by using also the rank as an invariant in some specific cases