3,099 research outputs found
A Deterministic Algorithm for Computing the Weight Distribution of Polar Codes
We present a deterministic algorithm for computing the entire weight
distribution of polar codes. As the first step, we derive an efficient
recursive procedure to compute the weight distributions that arise in
successive cancellation decoding of polar codes along any decoding path. This
solves the open problem recently posed by Polyanskaya, Davletshin, and
Polyanskii. Using this recursive procedure, we can compute the entire weight
distribution of certain polar cosets in time O(n^2). Any polar code can be
represented as a disjoint union of such cosets; moreover, this representation
extends to polar codes with dynamically frozen bits. This implies that our
methods can be also used to compute the weight distribution of polar codes with
CRC precoding, of polarization-adjusted convolutional (PAC) codes and, in fact,
general linear codes. However, the number of polar cosets in such
representation scales exponentially with a parameter introduced herein, which
we call the mixing factor. To reduce the exponential complexity of our
algorithm, we make use of the fact that polar codes have a large automorphism
group, which includes the lower-triangular affine group LTA(m,2). We prove that
LTA(m,2) acts transitively on certain sets of monomials, thereby drastically
reducing the number of polar cosets we need to evaluate. This complexity
reduction makes it possible to compute the weight distribution of any polar
code of length up to n=128
Families of nested completely regular codes and distance-regular graphs
In this paper infinite families of linear binary nested completely regular
codes are constructed. They have covering radius equal to or ,
and are -th parts, for of binary (respectively,
extended binary) Hamming codes of length (respectively, ), where
. In the usual way, i.e., as coset graphs, infinite families of embedded
distance-regular coset graphs of diameter equal to or are
constructed. In some cases, the constructed codes are also completely
transitive codes and the corresponding coset graphs are distance-transitive
Computing coset leaders and leader codewords of binary codes
In this paper we use the Gr\"obner representation of a binary linear code
to give efficient algorithms for computing the whole set of coset
leaders, denoted by and the set of leader codewords,
denoted by . The first algorithm could be adapted to
provide not only the Newton and the covering radius of but also to
determine the coset leader weight distribution. Moreover, providing the set of
leader codewords we have a test-set for decoding by a gradient-like decoding
algorithm. Another contribution of this article is the relation stablished
between zero neighbours and leader codewords
Explicit Constructions of Quasi-Uniform Codes from Groups
We address the question of constructing explicitly quasi-uniform codes from
groups. We determine the size of the codebook, the alphabet and the minimum
distance as a function of the corresponding group, both for abelian and some
nonabelian groups. Potentials applications comprise the design of almost affine
codes and non-linear network codes
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