The weight distribution of the self-dual $[128,64]$ polarity design code

The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design in $PG(6,2)$ [11] is computed. It is shown also that $R(3,7)$ and $C^{*}$ have no self-dual $[128,64,d]$ neighbor with $d \in \{ 20, 24 \}$.


Introduction
We assume familiarity with basic facts and notions from coding theory and combinatorial design theory ( [1,12,18]).
We denote by P G s (m, q) the design having as points and blocks the points and s-subspaces of the m-dimensional projective geometry P G(m, q) over a finite field GF (q) of order q, where q = p t is a prime power and 1 ≤ s ≤ m − 1. The projective geometry design P G s (m, q) is a 2-(v, k, λ) design with parameters where m i q denotes the Gaussian coefficient given by The affine geometry design AG s (m, q), (1 ≤ s ≤ m − 1), is a 2-(v, k, λ) design of the points and s-subspaces of the m-dimensional affine geometry AG(m, q) over GF (q), where In the special case when q = 2, AG s (m, 2), s ≥ 2, is also a 3-design, with every three points contained in λ 3 blocks, where A finite geometry code (or geometric code), is a linear code being the null space of the incidence matrix of a geometric design, AG s (m, q) or P G s (m, q). The codes based on affine geometry designs, AG s (m, q), are also called Euclidean geometry codes, while the codes based on P G s (m, q) are called projective geometry codes. The codes over GF (p) thus defined, where q = p t , correspond to subfield subcodes of generalized Reed-Muller codes [1, Chapter 5], [7]. The binary Euclidean geometry code being the null space of the incidence matrix of AG s (m, 2) is equivalent to the Reed-Muller code R(m − s, m) of length 2 m and order m − s. It is well known that the finite geometry codes admit majority logic decoding [8,16,19].
In [11], Jungnickel and Tonchev used polarities in projective geometry to find a class of designs which have the same parameters and share some other properties with a projective geometry design P G s (2s, q), s ≥ 2, but are not isomorphic to P G s (2s, q). We refer to these designs as polarity designs. In the cases when q = p is a prime, the p-rank of the incidence matrix of a polarity design D is equal to that of P G s (2s, p), hence the polarity designs provide an infinite class of counter-examples to Hamada's conjecture [9,10].
In [5], Clark and Tonchev proved that the code being the null space of the incidence matrix of a polarity design can correct by majority-logic decoding the same number of errors as the projective geometry code based on P G s (2s, q). In the binary case (q = 2), the minimum distance of the code of the polarity design obtained from P G(2s, 2), is equal to 2 s+1 , and the majority-logic algorithm from [5] corrects all errors guaranteed by the minimum distance. The extended code of the binary code spanned by the blocks of a polarity design obtained from P G(2s, 2) is a self-dual binary code of the same length, dimension and minimum distance, and correcting by majority-logic the same number of errors as the Reed-Muller code R(s, 2s + 1) of length 2 2s+1 and order s.
In the smallest binary case, s = 2, the extended code of the polarity design obtained from P G(4, 2), is a doubly-even self-dual [32, 16,8] code, which not only has the same parameters and corrects by majority-logic decoding the same number of errors as the 2nd order Reed-Muller code R(2, 5), but also has the same weight distribution as R (2,5). This phenomenon is easily explained by the fact that both codes are extremal doubly-even self-dual codes, hence are forced to have the same weight distribution [14] (actually, in this case there are five inequivalent extremal doubly-even self-dual [32, 16,8] codes [6].).
It is the aim of this note to report the computation of the weight distribution of the extended code of the polarity design in the next case s = 3, i.e. the polarity design obtained from P G(6, 2), and to demonstrate that this doubly-even self-dual [128, 64, 16] code has the same weight distribution as the 3rd order Reed-Muller code R(3, 7).
One of the authors, Vladimir Tonchev, conjectures that the extended code of the polarity design obtained from P G(2s, 2) has the same weight distribution as the Reed-Muller code R(s, 2s + 1) for every s ≥ 2.
The parameters and the block intersection numbers of D imply that the binary linear code C spanned by the block by point incidence matrix of D has minimum distance not exceeding 15, and its extended code C * is a doubly-even self-dual [128, 64] code of minimum distance d ≤ 16. It follows from the results from [4] and [5] that d = 16 and the code C * admits majority-logic decoding that corrects up to 7 errors, that is, the same number of errors as the doubly-even self-dual [128, 64, 16] 3rd order Reed-Muller code R(3, 7).
We will show that C * has the same weight distribution as R(3, 7). The weight distribution of the Reed-Muller code R(3, 7) was computed by Sugino, Ienaga, Tokura and Kasami [17] (see The On-line Encyclopedia of Integer Sequences [15], sequence A110845), and is listed in Table 1.
Since the code dimension 64 is significant, to facilitate the computation of the weight distribution of C * , we employ known properties of weight enumerators of binary doubly-even self-dual codes. Since C * is a doubly-even self-dual [128,64,16] code, by the Gleason theorem (cf. [13, Section 2], [14]), the weight distribution {A i } 128 i=0 of C * can be determined completely by the values of A 16 and A 20 . More specifically, the two-variable weight enumerator can be written as A 64 × 128 generator matrix G of the extended code C * was computed following the construction of polarity designs from [11], and is available online at http: Using Magma [2], it took a few minutes to compute on a PC that (4) A 16 = 94488, and about an hour 1 to compute Since the values A 16 (see (4)) and A 20 (see (5)) are the same as the corresponding values for the self-dual [128, 64, 16] Reed-Muller code R(3, 7) (cf.  16] code C * of the the code C spanned by the incidence vectors of the blocks of the polarity design D obtained from P G(6, 2), is identical with the weight distribution of the 3rd order Reed-Muller code R(3, 7).
Using Magma, it took 90 seconds to compute the full automorphism group Aut(C * ) of C * . Since C * is spanned by the set of minimum weight vectors which form the block by point incidence matrix of a 3-(128, 16, 155) design D * [4,5], the full automorphism group of C * coincides with that of D * , and is of order (6) | Aut(C * )| = 165140150353920 = 2 28 · 3 4 · 5 · 7 2 · 31.
Remark 1. The parameters of the extended self-dual code, obtained from the polarity design in P G (8,2), are [512,256,32]. It seems computationally infeasible to find the weight distribution of such a code by computer, even with the help of Gleason's theorem, due to the very large code dimension. Thus, any proof of the conjecture formulated in the last paragraph of Introduction, has to be based on geometric or other theoretical considerations.

Self-dual neighbors
In this section, we investigate self-dual neighbors of the 3rd order Reed-Muller code R(3, 7) and the extended [128, 64, 16] code C * given in Theorem 1. Two selfdual codes C and C of length n are said to be neighbors if dim(C ∩ C ) = n/2 − 1. We give some observations from [3] on self-dual codes constructed by neighbors. Let C be a self-dual [n, n/2, d] code. Let M be a matrix whose rows are the codewords of weight d in C. Suppose that there is a vector x ∈ F n 2 such that (8) where x T denotes the transpose of x and 1 is the all-one vector. Set C 0 = x ⊥ ∩ C, where x denotes the code generated by x. Then C 0 is a subcode of index 2 in C. If the weight of x is even, then we have the two self-dual neighbors C 0 , x and C 0 , x + y of C for some y ∈ C \ C 0 , which do not have any codeword of weight d in C, where C, x = C ∪ (x + C). This means that the above two doubly-even self-dual codes of length 128 have no extremal doubly-even self-dual neighbor of that length.