1. Construction of a Self-Dual [94,47,16] Code
Masaaki Harada
Department of Mathematical Sciences
Yamagata University, Japan
Radinka Yorgova
Department of Informatics
University of Bergen, Norway

2. 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

3. • 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

4. if 24/n ) 8 code meeting the bound must be doubly-even
, if n ´ 22 (mod 24)
if 24/n ) 8 code meeting the bound must be doubly-even
and
, if n ´ 22 (mod 24)
self-dual codes reaching these bounds are called extremal
C, C’ - binary [n,k] codes
C ~ C’ (equivalent) if 9  2 Sn : C’ = (C) ,
if C = (C) for some  2 Sn )  is called an automorphism of C

5. An extremal doubly even self-dual [24k,12k,4k+4] code is known for only k=1,2
- the extended Golay [24,12,8] code
- the extended quadratic residue [48,24,12] code
? for k>2
The existences of an extremal doubly even self-dual [24k, 12k, 4k+4] code is equivalent to
the existences a self-dual [24k-2, 12k-1, 4k+2] code.
? the largest d among self-dual codes of length 24k-2
n= d=12 or 14 d=14 - an open case
n= d=14, 16 or 18 d=16, open cases
Here we construct a self-dual [94,47,16] code for the first time
) n= d=16 or 18 d=18 - an open case

6. We use the well known method developed by Huffman in 1982
• 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.

7.  - an automorphism of C94 , | | =23
C94 - a [94,47,16] self-dual code
 - an automorphism of C94 , | | =23
94 =
ei , fi – 11£ 23 right circulant matrices

8. ei , fi – 11£ 23 right circulant matrices
the first rows of ei , fi correspond to polynomials in F2 [x] / (x23 - 1)
e1  e(x) = x22+x21+x20+x19+x17+x15+x14+x11+x10+x7+x5+1
e2 ®1(x)=x20+x17+x15+x14+x13+x12+x11+x10+x7+x3+x+1
e3 ®136(x), e4 x11®113(x)
f1  e(x -1), f2 ®1(x-1), f3 ®136(x-1), f4 x12®113(x-1)

9. 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.

10. 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

11. Theorem (J.H.Conway and N.J.A.Sloane)
Let S=S(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 u+v 2 C(0);
if u 2 C(1), v 2 C(3) then u+v 2 C(2) ;
and if u,v 2 C(3) then u+v 2 C(0)

12. 3) Let be the weight enumerator of S
Then
where W(x; y) is the weight enumerator of C, Also for all r,
, for r 6´ n=2 (mod 4), ,
, for r < d=2,
, for all r, and at most one is nonzero for r < (d + 4)=2, where
A(n; d; r) denotes the maximal possible number of binary vectors of length n, weight r
and Hamming distance at least d.

13. The weight enumerators of C94 and S94 a are:
WC = ® y16+ ( ® ¯ ) y18+ ( ® ¯ ° )y20+
( ® ¯ ° ± ) 22+ 
WS = ± y3 +(° - 22 ± ) y7 +(- ¯ ° ± ) y11 +(® + 18 ¯ ° ± ) y15 +
( ® ¯ ° ± ) y19 + 
By 3) of the Theorem ) (± ,° ) = (0,0), (0,1), (1,22)
For our code d(S)=15 and A16= ) (®, ¯, °, ±) = (3036, 0, 0, 0)

14. • 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 n+2
(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