Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn two-weight codes
120
0
0.0
(
0
)
Added by
Peter Boyvalenkov
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We consider $q$-ary (linear and nonlinear) block codes with exactly two distances: $d$ and $d+delta$. Several combinatorial constructions of optimal such codes are given. In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear $2$-weight code with $delta > 1$ implies the following equality of great common divisors: $(d,q) = (delta,q)$. Upper bounds for the maximum cardinality of such codes are derived by linear programming and from few-distance spherical codes. Tables of lower and upper bounds for small $q = 2,3,4$ and $q,n < 50$ are presented.
rate research
Read More
The determination of weight distribution of cyclic codes involves evaluation of Gauss sums and exponential sums. Despite of some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectrums of Hermitian forms graphs.
In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $Tr(x^d)$ over $F_{2^{m}}$ in the case where $m=2t$, $d=3+2^{t+1}$ and $gcd(d, 2^{m}-1)=1$.
Let $p$ be a prime number. Irreducible cyclic codes of length $p^2-1$ and dimension $2$ over the integers modulo $p^h$ are shown to have exactly two nonzero Hamming weights. The construction uses the Galois ring of characteristic $p^h$ and order $p^{2h}.$ When the check polynomial is primitive, the code meets the Griesmer bound of (Shiromoto, Storme) (2012). By puncturing some projective codes are constructed. Those in length $p+1$ meet a Singleton-like bound of (Shiromoto , 2000). An infinite family of strongly regular graphs is constructed as coset graphs of the duals of these projective codes. A common cover of all these graphs, for fixed $p$, is provided by considering the Hensel lifting of these cyclic codes over the $p$-adic numbers.
CSS codes are in one-to-one correspondance with length 3 chain complexes. The latter are naturally endowed with a tensor product $otimes$ which induces a similar operation on the former. We investigate this operation, and in particular its behavior with regard to minimum distances. Given a CSS code $mathcal{C}$, we give a criterion which provides a lower bound on the minimum distance of $mathcal{C} otimes mathcal{D}$ for every CSS code $mathcal D$. We apply this result to study the behaviour of iterated tensor powers of codes. Such sequences of codes are logarithmically LDPC and we prove in particular that their minimum distances tend generically to infinity. Different known results are reinterpretated in terms of tensor products. Three new families of CSS codes are defined, and their iterated tensor powers produce LDPC sequences of codes with length $n$, row weight in $O(log n)$ and minimum distances larger than $n^{frac{alpha}{2}}$ for any $alpha<1$. One family produces sequences with dimensions larger than $n^beta$ for any $beta<1$.
We determine the proportion of $[3times 3;3]$-MRD codes over ${mathbb F}_q$ within the space of all $3$-dimensional $3times3$-rank-metric codes over the same field. This shows that for these parameters MRD codes are sparse in the sense that the proportion tends to $0$ as $qrightarrowinfty$. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
