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SYMMETRY CODES AND THEIR INVARIANT SUBCODES

IP.com Disclosure Number: IPCOM000149195D
Original Publication Date: 1899-Dec-30
Included in the Prior Art Database: 2007-Apr-12

Publishing Venue

Software Patent Institute

Related People

Pless, Vera: AUTHOR [+2]

Abstract

SYMMETRY CODES AND THEIR IPJVaUR'hANT SUBCODES T h i s research was s u p p o r t e d by k h e Advanced R e s e a r c h Projects Agency sf the D e p a r t m e n t of D e f e n s e u n d e r _UP& O r d e r No. 2095, and was m o n i t o r e d by ONR under C o n t r a c t No. N00014-70-A-8362-0006. WSACHUSETTS INSTITUTE OF TECHNOLOGY PROJECT MA.6: MASSACHUSETTS 02139 Vera Pless CAMBRIDGE Symmetry Codes and Their 1nva:riant Subcodes Abstract We define and study the invariant subcodes of the symmetry codes i n order t o be able t o determine the algebraic properties of these codes. An infinite farniky of self-orthogonal rate 112 codes over GF(3), called symmetry codes, were constructed i n 3 . A (2q -e 2, q c 1) symmetry code , denoted by C(q) , exists whenever q is an odd prime power -1, (mod 3). The group of monomial transfor- mations leaving a symmetry code invariant is denoted by G(q). En this paper we construct two subcodes of C(q) denoted by R (q) and R (q). 0 P

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SYMMETRY CODES AND THEIR IPJVaUR'hANT SUBCODES

Vera Pless

T h i s research was s u p p o r t e d by k h e Advanced R e s e a r c h Projects Agency sf the D e p a r t m e n t of D e f e n s e u n d e r _UP& O r d e r No. 2095, and was m o n i t o r e d by ONR under C o n t r a c t No. N00014-70-A-8362-0006.

WSACHUSETTS INSTITUTE OF TECHNOLOGY

PROJECT MA.6:

MASSACHUSETTS 02139

CAMBRIDGE

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Symmetry Codes and Their 1nva:riant Subcodes

Abstract

         We define and study the invariant subcodes of the symmetry codes i n order t o be able t o determine the algebraic properties of these codes. An infinite farniky of self-orthogonal rate 112 codes over GF(3), called symmetry codes, were constructed i n [ 3 ] . A
(2q -e 2, q c 1) symmetry code , denoted by C(q) , exists whenever q
is an odd prime power = -1, (mod 3). The group of monomial transfor- mations leaving a symmetry code invariant is denoted by G(q). En this paper we construct two subcodes of C(q) denoted by R (q) and R (q).

                                                                  0 P
Every vector i n R (q) is invariant under a monomial transformation 7

(5

i n G(q) of odd order s where s divides (q f 1). Also R (q) is invariant

P

under T but not vector-wise. The dimensions of K (q) and R (q) are de-

                                                           O P
termined and relations between these subcodes are given. An isomorphism

232L

is constructed between %a (q) and a subspace of W = V3 s It is

0

shown that the image of R (q) is a self-orthogonal subspace of W. The

0

isomorphic images of R (17) (under an order 3 monomial) and R (29)

0- 0

(under an order 5 monomial) are both demonstrated to be equivalent t o the (12, 6) Golay code.

D r . Vera Pless


Pro j ect MAC
MassachuseQts Institute of Technology 545 Technology Square, Rm. 530 Cambridge, Massachusetts 02139

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Symmetry Codes and Their Invariant Subcodes by
D r . Vera Pless

Proj e c M C

I. Introduction.

This paper defines and studies the invariant subcodes of the

symmetry codes which were originally defined i n [3]. The purpose of this study is the illucidation of properties of these subcodes i n such a manner that these properties can be applied i n determining character- istics of the symmetry code itself. For example, maximum length vec- tors i n C(17) and C(29) can. be determined from known maximum length vectors i n the Golay cod$? C(5). The minimum weights are known for the f i r s t five symmetry codes. Estimates of the minimum weights of the larger symmetry codes have been obtained by locating a vector of weight 21 i n R. (41) (under an order 7 monomial) and a vector of weight 27 i n

0

R (53) (under an order 3 monomial), An (n, k) error correcting code 0
n

ove...