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[0].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