# Enumerating binary self-dual codes¶

This module implements functions useful for studying binary self-dual codes. The main function is self_dual_binary_codes, which is a case-by-case list of entries, each represented by a Python dictionary.

Format of each entry: a Python dictionary with keys “order autgp”, “spectrum”, “code”, “Comment”, “Type”, where

• “code” - a sd code C of length n, dim n/2, over GF(2)

• “order autgp” - order of the permutation automorphism group of C

• “Type” - the type of C (which can be “I” or “II”, in the binary case)

• “spectrum” - the spectrum [A0,A1,…,An]

• “Comment” - possibly an empty string.

Python dictionaries were used since they seemed to be both human-readable and allow others to update the database easiest.

• The following double for loop can be time-consuming but should be run once in awhile for testing purposes. It should only print True and have no trace-back errors:

for n in [4,6,8,10,12,14,16,18,20,22]:
C = self_dual_binary_codes(n); m = len(C.keys())
for i in range(m):
C0 = C["%s"%n]["%s"%i]["code"]
print([n,i,C["%s"%n]["%s"%i]["spectrum"] == C0.spectrum()])
print(C0 == C0.dual_code())
G = C0.automorphism_group_binary_code()
print(C["%s" % n]["%s" % i]["order autgp"] == G.order())

• To check if the “Riemann hypothesis” holds, run the following code:

R = PolynomialRing(CC,"T")
T = R.gen()
for n in [4,6,8,10,12,14,16,18,20,22]:
C = self_dual_binary_codes(n); m = len(C["%s"%n].keys())
for i in range(m):
C0 = C["%s"%n]["%s"%i]["code"]
if C0.minimum_distance()>2:
f = R(C0.sd_zeta_polynomial())
print([n,i,[z.abs() for z in f.roots()]])


You should get lists of numbers equal to 0.707106781186548.

Here’s a rather naive construction of self-dual codes in the binary case:

For even m, let A_m denote the mxm matrix over GF(2) given by adding the all 1’s matrix to the identity matrix (in MatrixSpace(GF(2),m,m) of course). If M_1, …, M_r are square matrices, let $$diag(M_1,M_2,...,M_r)$$ denote the”block diagonal” matrix with the $$M_i$$ ‘s on the diagonal and 0’s elsewhere. Let $$C(m_1,...,m_r,s)$$ denote the linear code with generator matrix having block form $$G = (I, A)$$, where $$A = diag(A_{m_1},A_{m_2},...,A_{m_r},I_s)$$, for some (even) $$m_i$$ ‘s and $$s$$, where $$m_1+m_2+...+m_r+s=n/2$$. Note: Such codes $$C(m_1,...,m_r,s)$$ are SD.

SD codes not of this form will be called (for the purpose of documenting the code below) “exceptional”. Except when n is “small”, most sd codes are exceptional (based on a counting argument and table 9.1 in the Huffman+Pless [HP2003], page 347).

AUTHORS:

• David Joyner (2007-08-11)

REFERENCES:

• [HP2003] W. C. Huffman, V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003.

• [P] V. Pless, “A classification of self-orthogonal codes over GF(2)”, Discrete Math 3 (1972) 209-246.

sage.coding.self_dual_codes.self_dual_binary_codes(n)

Returns the dictionary of inequivalent binary self dual codes of length n.

For n=4 even, returns the sd codes of a given length, up to (perm) equivalence, the (perm) aut gp, and the type.

The number of inequiv “diagonal” sd binary codes in the database of length n is (“diagonal” is defined by the conjecture above) is the same as the restricted partition number of n, where only integers from the set 1,4,6,8,… are allowed. This is the coefficient of $$x^n$$ in the series expansion $$(1-x)^{-1}\prod_{2^\infty (1-x^{2j})^{-1}}$$. Typing the command f = (1-x)(-1)*prod([(1-x(2*j))(-1) for j in range(2,18)]) into Sage, we obtain for the coeffs of $$x^4$$, $$x^6$$, … [1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 11, 11, 15, 15, 22, 22, 30, 30, 42, 42, 56, 56, 77, 77, 101, 101, 135, 135, 176, 176, 231] These numbers grow too slowly to account for all the sd codes (see Huffman+Pless’ Table 9.1, referenced above). In fact, in Table 9.10 of [HP2003], the number B_n of inequivalent sd binary codes of length n is given:

n   2 4 6 8 10 12 14 16 18 20 22 24  26  28  30
B_n 1 1 1 2  2  3  4  7  9 16 25 55 103 261 731


According to http://oeis.org/classic/A003179, the next 2 entries are: 3295, 24147.

EXAMPLES:

sage: C = codes.databases.self_dual_binary_codes(10)
sage: C["10"]["0"]["code"] == C["10"]["0"]["code"].dual_code()
True
sage: C["10"]["1"]["code"] == C["10"]["1"]["code"].dual_code()
True
sage: len(C["10"].keys()) # number of inequiv sd codes of length 10
2
sage: C = codes.databases.self_dual_binary_codes(12)
sage: C["12"]["0"]["code"] == C["12"]["0"]["code"].dual_code()
True
sage: C["12"]["1"]["code"] == C["12"]["1"]["code"].dual_code()
True
sage: C["12"]["2"]["code"] == C["12"]["2"]["code"].dual_code()
True