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A New Construction for the Extended Binary Golay Code
Abstract
We give a new construction of the extended binary Golay code. The construction is carried out by taking the Gray image of a self-dual linear code over the ring R = F+2 + uF2 + vF2 + uvF2 of length 6 and size 212.Writing a typical generating matrix of the form [I3|A], with A being a 3×3 matrix over R, and finding some dependencies among the entries of A, we are able to set a general form for the generating matrices of self-dual codes of length 6. Using some special properties of elements of R, we end up with a family of generating matrices all of which give us the extended binary Golay code. We also prove the minimum distance property analytically.
... In [13], the authors find a Type II self-dual code over F 2 [u, v]/ u 2 , v 2 whose image under φ is the extended binary Golay code. ...
We study codes over the commutative local Frobenius rings of order 16 with maximal ideals of size 8. We define a weight preserving Gray map and study the images of these codes as binary codes. We study self-dual codes and determine when they exist.
... In [13] ...
Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators can be easily found. As examples, generating characters for all commutative local Frobenius rings of order 16 are given. In addition, a canonical generator matrix for codes over local non-chain rings is discussed. The purpose is to show that when working over local non-chain rings, a canonical generator matrix exists but is less than useful which emphases the difficulties in working over such rings.
... [19], [22] are examples of these constructions. In [14], the extended Golay code was constructed from what we now call R 2,2 . ...
In this work, we study codes over the ring
R_{k,m}=F_2[u,v]/<u^{k},v^{m},uv-vu>, which is a family of Frobenius,
characteristic 2 extensions of the binary field. We introduce a distance and
duality preserving Gray map from R_{k,m} to F_2^{km} together with a Lee
weight. After proving the MacWilliams identities for codes over R_{k,m} for all
the relevant weight enumerators, we construct many binary self-dual codes as
the Gray images of self-dual codes over R_{k,m}. In addition to many extremal
binary self-dual codes obtained in this way, including a new construction for
the extended binary Golay code, we find 175 new Type I binary self-dual codes
of parameters [72,36,12] and 105 new Type II binary self-dual codes of
parameter [72,36,12].
In this work, we study codes over the ring 𝓡k, m = 𝔽2[u, v]/ 〈uk, vm, uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡k, m to F2km together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡k, m. In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].
Two product array codes are used to construct the (24,12,8) binary Golay code through the direct sum operation. This construction provides a systematic way to find proper (8,4,4) linear block component codes for generating the Golay code, and it generates and extends previously existing methods that use a similar construction framework. The code constructed is simple to decode
The alphabet F2+uF2 is viewed here as a
quotient of the Gaussian integers by the ideal (2). Self-dual F<sub>2
</sub>+uF2 codes with Lee weights a multiple of 4 are called
Type II. They give even unimodular Gaussian lattices by Construction A,
while Type I codes yield unimodular Gaussian lattices. Construction B
makes it possible to realize the Leech lattice as a Gaussian lattice.
There is a Gray map which maps Type II codes into Type II binary codes
with a fixed point free involution in their automorphism group.
Combinatorial constructions use weighing matrices and strongly regular
graphs. Gleason-type theorems for the symmetrized weight enumerators of
Type II codes are derived. All self-dual codes are classified for length
up to 8. The shadow of the Type I codes yields bounds on the highest
minimum Hamming and Lee weights
Type II Z 4-codes are introduced as self-dual
codes over the integers modulo 4 containing the all-one vector and with
Euclidean weights multiple of 8. Their weight enumerators are
characterized by means of invariant theory. A notion of extremality for
the Euclidean weight is introduced. Their binary images under the Gray
map are formally self-dual with even weights. Extended quadratic residue
Z 4-codes are the main example of this family of
codes. They are obtained by Hensel lifting of the classical binary
quadratic residue codes. Their binary images have good parameters. With
every type II Z 4-code is associated via construction
A modulo 4 an even unimodular lattice (type II lattice). In dimension
32, we construct two unimodular lattices of norm 4 with an automorphism
of order 31. One of them is the Barnes-Wall lattice BW32
It is shown that the minimal distance d of a binary
self-dual code of length n ⩾74 is at most
2[( n +6)/10]. This bound is a consequence of some new conditions
on the weight enumerator of a self-dual code obtained by considering a
particular translate of the code, called its shadow. These conditions
also enable one to find the highest possible minimal distance of a
self-dual code for all n ⩾60; to show that self-dual codes
with d ⩽6 exist precisely for n ⩾22, with d
⩾8 exist precisely for n =24, 32 and n ⩾26,
and with d ⩾10 exist precisely for n ⩾46; and to
show that there are exactly eight self-dual codes of length 32 with
d =8. Several of the self-dual codes of length 34 have trivial
group (this appears to be the smallest length where this can happen)
The focus in this work is on self-dual codes over the ring F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2. Type I and Type II codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 are defined and some of the techniques used in the literature are applied to get some theoretical results about self-dual codes on F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2 and their binary images. In particular some extremal and optimal binary codes of Type I and Type II including the extended Golay code are obtained as the Gray images of codes over F2+uF2+vF2+uvF2F2+uF2+vF2+uvF2.
In this work, we investigate linear codes over the ring
. We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43–65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32–45, 1999). We then characterize the
-linearity of binary codes under the Gray map and give a main class of binary codes as an example of
-linear codes. The duals and the complete weight enumerators for
-linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over
are obtained.
We give a new construction of the extended binary Golay code. The code is constructed as a zero divisor code over the dihedral group of twenty-four elements, D 24. The construction is algebraic, much like the construction of the (23,12,7) binary Golay code from a polynomial.