A076766 Number of inequivalent binary linear codes of length n. Also the number of nonisomorphic binary matroids on an n-set.

1, 2, 4, 8, 16, 32, 68, 148, 342, 848, 2297, 6928, 24034, 98854, 503137, 3318732, 29708814, 374039266, 6739630253, 173801649708, 6356255181216, 326203517516704, 23294352980140884, 2301176047764925736, 313285408199180770635, 58638266023262502962716

Offset: 0

Author

Marcel Wild (mwild(AT)sun.ac.za), Nov 14 2002

Keywords

• nice
• nonn

Examples

			a(2)=4 because there are four inequivalent linear binary 2-codes: {(0,0)}, {(0,0),(1,0)}, {(0,0),(1,1)}, {(0,0),(1,0),(0,1),(1,1)}. Observe that the codes {(0,0),(1,0)} and {(0,0),(0,1)} are equivalent because one arises from the other by a permutation of coordinates.
		

References

• M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint Nr. 1693, Tech. Hochschule Darmstadt, 1994.

Links

• Jayant Apte and J. M. Walsh, Constrained Linear Representability of Polymatroids and Algorithms for Computing Achievability Proofs in Network Coding, arXiv preprint arXiv:1605.04598 [cs.IT], 2016-2017.
• Suyoung Choi and Hanchul Park, Multiplication structure of the cohomology ring of real toric spaces, arXiv:1711.04983 [math.AT], 2017.
• H. Fripertinger, Isometry Classes of Codes.
• Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007.
• Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
• James Oxley, What is a Matroid?.
• Gordon Royle and Dillon Mayhew, 9-element matroids.
• D. Slepian, On the number of symmetry types of Boolean functions of n variables, Canadian J. Math. 5, (1953), 185-193.
• D. Slepian, A class of binary signaling alphabets, Bell System Tech. J. 35 (1956), 203-234.
• D. Slepian, Some further theory of group codes, Bell System Tech. J. 39 1960 1219-1252. (Row sums of Table II.)
• Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316.
• Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202.
• Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM J. Discrete Math. 19 (2005), 691-699. [This paper apparently corrects some errors in previous papers.]
• Index entries for sequences related to binary linear codes

Crossrefs

Row sums of triangle A076831. Cf. A034328, A055545.

Extensions

Edited by N. J. A. Sloane, Nov 01 2007, at the suggestion of Gordon Royle.