A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field was recently presented in [10]. Shortly after this, a generalization for the sufficient numerical conditions of such characterization was given in [3]. The main purpose of this work is to show that the numerical conditions found in [3], are also necessary. As we will see later, an interesting...

In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in Z with radius r ≥ 2 and dimension n ≥ 3. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this ...

Approximately dual frames are studied in the Hilbert space setting. Approximate duals are easier to construct than classical dual frames, and can be tailored to yield almost perfect reconstruction. Bounds on the deviation from perfect reconstruction are obtained for approximately dual frames constructed via perturbation theory. An alternative bound is derived for the rich class of Gabor frames,...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.

We establish a characterization of the extraordinary dimension of perfect maps between metrizable spaces.