Construction%20of%20a%20Self-Dual%20[94,47,16]%20Code

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Construction of a Self-Dual 94,47,16 Code

• Radinka Yorgova
• Department of Informatics
• University of Bergen, Norway

Masaaki Harada Department of Mathematical
Sciences Yamagata University, Japan
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Outline notations a self-dual
94,47,16 code having an automorphism of order
23 weight enumerators of self-dual
94,47,16 codes a related self-dual code of
length 96
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notations
F2 the binary field
F2n- the standard vector space over F2
v2 F2n , wt(v) i vi ? 0, 1 i n
Hamming weight of v
a binary n, k, d code C - k-dimensional
subspace of F2n, where d min wt(v) v
2 C, v? 0
G - a generator matrix of C, n x k matrix
C? - the dual code of C under the standard
inner product
C self-dual code if C C?
if C self-dual ) k n/2 and 2/wt(v), 8 v
2 C
if 4/wt(v), 8 v 2 C ) C is doubly-even,
otherwise C is singly-even
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and
self-dual codes reaching these bounds are called
extremal
if C ?(C) for some ? 2 Sn ) ? is called an
automorphism of C
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An extremal doubly even self-dual 24k,12k,4k4
code is known for only k1,2 - the extended
Golay 24,12,8 code - the extended quadratic
residue 48,24,12 code ? for kgt2 The
existences of an extremal doubly even self-dual
24k, 12k, 4k4 code is equivalent to the
existences a self-dual 24k-2, 12k-1, 4k2
code. ? the largest d among self-dual codes of
length 24k-2 n70 d12 or 14 d14 - an
open case n94 d14, 16 or 18 d16, 18 -
open cases Here we construct a self-dual
94,47,16 code for the first time ) n94
d16 or 18 d18 - an open case
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a self-dual 94,47,16 code having an
automorphism of order 23
We use the well known method developed by
Huffman in 1982 and improved by Yorgov in 1983
for constructing binary-self dual codes under
the assumption that the codes have an
automorphism of given odd prime order.
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C94 - a 94,47,16 self-dual code
? - an automorphism of C94 , ? 23
94 4 .23 2
ei , fi 11 23 right circulant matrices
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ei , fi 11 23 right circulant matrices
the first rows of ei , fi correspond to
polynomials in F2 x / (x23 - 1) e1 ? e(x)
x22x21x20x19x17x15x14x11x10x7x51
e2? 1(x)x20x17x15x14x13x12x11x10x7x3
x1 e3? 136(x), e4? x11113(x) f1 ? e(x
-1), f2? 1(x-1), f3? 136(x-1), f4?
x12113(x-1)
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Proposition 1 There is a self-dual 94,47,16
code. The largest minimum weight among self-dual
codes of length 94 is 16 or 18.
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Shadow of a self-dual code C - a singly even
self-dual code C(0) c 2 C wt(c) 0 (mod
4) C(2) C \ C(0) The shadow S(C) parity
vectors u for C u . v 0 for all v 2
C(0) u . v 1 for all v 2 C(2) If C
is a code of Type II, then C(0) C, C(2)
and S(C ) C
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Theorem (J.H.Conway and N.J.A.Sloane) Let SS(C)
be the shadow code corresponding to an n, n/2,
d Type I self-dual code C. The dual C(0)? C(0)
C(1) C(2) C(3) with C C(0) C(2) . 1)
S C(0)? \ C C(1) C(3) 2) The sum of any
two vectors in S is in C. More precisely, if
u,v 2 C(1) then uv 2 C(0) if u 2 C(1), v 2
C(3) then uv 2 C(2) and if u,v 2 C(3) then
uv 2 C(0)
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(No Transcript)
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The weight enumerators of C94 and S94 a are
WC 1 2 y16 (134044 - 2 128 ) y18
(2010660 - 30 - 896 8192 )y20
(22385348 - 30 -1280 - 106496 - 524288 )
22 ? WS y3 ( - 22 ) y7 (- - 20
231 ) y11 ( 18 190 - 1540 )
y15 (1072352 - 16 - 153 -
1140 7315 ) y19 ?
By 3) of the Theorem ) ( , ) (0,0), (0,1),
(1,22)
For our code d(S)15 and A16 6072 ) (, ,
, ) (3036, 0, 0, 0)
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a related self-dual code of length 96 Let C
be a Type I self-dual code of length n 6 (mod
8). Let C be the code of length n2 (0, 0,
C(0) ) (1, 0, C(1) ) (1, 1, C(2) ) (0, 1,
C(3) ) Where C(0)? C(0) C(1) C(2) C(3),
C C(0) C(2) S
C(1) C(3) Then C is a doubly even
self-dual code (1991, R.Brualdi and V.Pless) In
our case C is a self-dual 94,47,16 code with
shadow S of weight 15, then C is a doubly even
self dual 96,48,16 code