97 research outputs found
On the Residue Codes of Extremal Type II Z4-Codes of Lengths 32 and 40
In this paper, we determine the dimensions of the residue codes of extremal
Type II Z4-codes for lengths 32 and 40. We demonstrate that every binary doubly
even self-dual code of length 32 can be realized as the residue code of some
extremal Type II Z4-code. It is also shown that there is a unique extremal Type
II Z4-code of length 32 whose residue code has the smallest dimension 6 up to
equivalence. As a consequence, many new extremal Type II Z4-codes of lengths 32
and 40 are constructed.Comment: 19 page
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
Residue codes of extremal Type II Z_4-codes and the moonshine vertex operator algebra
In this paper, we study the residue codes of extremal Type II Z_4-codes of
length 24 and their relations to the famous moonshine vertex operator algebra.
The main result is a complete classification of all residue codes of extremal
Type II Z_4-codes of length 24. Some corresponding results associated to the
moonshine vertex operator algebra are also discussed.Comment: 21 pages, shortened from v
A new class of codes for Boolean masking of cryptographic computations
We introduce a new class of rate one-half binary codes: {\bf complementary
information set codes.} A binary linear code of length and dimension
is called a complementary information set code (CIS code for short) if it has
two disjoint information sets. This class of codes contains self-dual codes as
a subclass. It is connected to graph correlation immune Boolean functions of
use in the security of hardware implementations of cryptographic primitives.
Such codes permit to improve the cost of masking cryptographic algorithms
against side channel attacks. In this paper we investigate this new class of
codes: we give optimal or best known CIS codes of length We derive
general constructions based on cyclic codes and on double circulant codes. We
derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all
be classified in small lengths by the building up construction. Some
nonlinear permutations are constructed by using -codes, based on the
notion of dual distance of an unrestricted code.Comment: 19 pages. IEEE Trans. on Information Theory, to appea
On some new extremal Type II Z4-codes of length 40
Using the building-up method and a modification of the doubling method we construct new extremal Type II Z4-codes of length 40. The constructed codes of type , for ,
are the first examples of extremal Type II Z4-codes of given type and length 40 whose residue codes have minimum weight greater than or equal to 8. Further, we use minimum weight codewords for a construction of 1-designs, some of which are self-orthogonal
New extremal Type II -codes of length 64 by the doubling method
Extremal Type II -codes are a class of self-dual
-codes with Euclidean weights divisible by eight and the largest
possible minimum Euclidean weight for a given length. A small number of such
codes is known for lengths greater than or equal to The doubling method
is a method for constructing Type II -codes from a given Type II
-code. Based on the doubling method, in this paper we develop a
method to construct new extremal Type II -codes starting from an
extremal Type II -code of type with an extremal residue
code and length or . Using this method, we construct three new
extremal Type II -codes of length and type .
Extremal Type II -codes of length of this type were not
known before. Moreover, the residue codes of the constructed extremal
-codes are new best known binary codes and the supports
of the minimum weight codewords of the residue code and the torsion code of one
of these codes form self-orthogonal -designs
- …