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

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

• 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

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

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=70 d=12 or 14 d=14 - an open case n=94 d=14, 16 or 18 d=16, 18 - open cases Here we construct a self-dual [94,47,16] code for the first time ) n=94 d=16 or 18 d=18 - an open case

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.

 - an automorphism of C94 , | | =23 C94 - a [94,47,16] self-dual code  - an automorphism of C94 , | | =23 94 = 4 .23 + 2 ei , fi – 11£ 23 right circulant matrices

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)

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.

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

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)

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.

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)

• 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