A105674 -id:A105674 - cp's OEIS frontend

A066016 Highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 5, 6, 7, 8

Offset: 1

N. J. A. Sloane, Dec 12 2001; revised May 06 2005

S. T. Dougherty, M. Harada and P. Solé, Shadow Codes over Z_4, Finite Fields Applic., 7 (2001), 507-529.
P. Gaborit, Tables of Self-Dual Codes
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105677, A105678, A016729, A105681, A105682.

Cf. A066017 for number of codes. See also A066012-A066017, A098068.

A105676 Highest minimal Hamming distance of any Type 3 ternary self-dual code of length 4n.

Original entry on oeis.org

3, 3, 6, 6, 6, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18

Offset: 1

N. J. A. Sloane, May 06 2005

			The [12,6,6]_3 ternary Golay code has d=6, so a(3) = 6.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

P. Gaborit, Tables of Self-Dual Codes
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105677, A105678, A016729, A066016, A105681, A105682.

Cf. also A105683.

The sequence continues: a(17) = either 15 or 18, a(18) = 18, ...

A105682 Highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z.

Original entry on oeis.org

4, 4, 4, 4, 4, 4, 4, 8, 4, 4, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 12, 16

Offset: 1

N. J. A. Sloane, May 06 2005

S. T. Dougherty, M. Harada and P. Solé, Shadow Codes over Z_4, Finite Fields Applic., 7 (2001), 507-529.
P. Gaborit, Tables of Self-Dual Codes
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681.

A105689 gives number of such codes. Cf. also A066014.

A016729 Highest minimal Hamming distance of any Type 4^H+ Hermitian additive self-dual code over GF(4) of length n.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 3, 4, 4, 4, 5, 6, 5, 6, 6, 6, 7, 8, 7, 8, 8, 8

Offset: 1

N. J. A. Sloane. Entry revised May 06 2005

The sequence continues: a(23) = 8 or 9, a(24) = 8, 9 or 10, a(25) = 8 or 9, ...

P. Gaborit and A. Otmani, Experimental construction of self-dual codes, Preprint.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
P. Gaborit, Tables of Self-Dual Codes
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105677, A105678, A066016, A105681, A105682.

A105687 gives the number of codes with this minimal distance.

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n > 7.
G.f.: x*(-2*x^6 + x^5 + x^4 + x + 1)/(x^7 - x^6 - x + 1). (End)

A105675 Highest minimal distance of any Type II doubly-even binary self-dual code of length 8n.

Original entry on oeis.org

4, 4, 8, 8, 8, 12, 12, 12

Offset: 1

N. J. A. Sloane, May 06 2005

Is a(9) = 12 or 16? This is an open question of long standing.

			At length 8 the only Type II doubly-even self-dual code is the Hamming code e_8, which has d=4, so a(1) = 4. The [24,12,8] Golay code has d=8, so a(3) = 8.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
P. Gaborit, Tables of Self-Dual Codes
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).
N. J. A. Sloane, Is There a (72,36) d = 16 Self-Dual Code?, IEEE Trans. Information Theory, vol. IT-19 (1973), p. 251.

Cf. A105674, A105676, A105677, A105678, A016729, A066016, A105681, A105682.

Cf. also A001380, A018236.

A105677 Highest minimal Hamming distance of any Type 4^E Euclidean linear self-dual code over GF(4) of length 2n.

Original entry on oeis.org

2, 3, 3, 4, 4, 6, 6, 6

Offset: 1

N. J. A. Sloane, May 06 2005

There is a related sequence which is presently too short to include: Highest minimal Lee distance of any Type (4^E)_II Euclidean linear even self-dual code over GF(4) of length 4n. This begins 4, 4, 8, 8, 8, then either 8 or 12, 12, 12, ...

The sequence continues: a(9) = either 6 or 7, a(10) = a(11) = 8, a(12) = 8, 9 or 10, ...

P. Gaborit and A. Otmani, Experimental construction of self-dual codes, Preprint.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
P. Gaborit, Tables of Self-Dual Codes
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105678, A016729, A066016, A105681, A105682.

A105678 Highest minimal Hamming distance of any Type 4^H Hermitian linear self-dual code over GF(4) of length 2n.

Original entry on oeis.org

2, 2, 4, 4, 4, 4, 6, 6, 8, 8, 8, 8, 8, 10, 12

Offset: 1

N. J. A. Sloane, May 06 2005

The next term a(16) is either 10 or 12.

P. Gaborit, Tables of Self-Dual Codes
P. Gaborit and A. Otmani, Experimental construction of self-dual codes, Finite Fields and Their Applications, Volume 9, Issue 3, July 2003, Pages 372-394.
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105677, A016729, A066016, A105681, A105682.

Cf. also A105686 for the numbers of such codes.

A105681 Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z.

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 4, 6, 2, 4, 4, 4, 4, 6, 6, 8, 6, 8, 6, 8, 8, 8, 10, 12

Offset: 1

N. J. A. Sloane, May 06 2005

S. T. Dougherty, M. Harada and P. Solé, Shadow Codes over Z_4, Finite Fields Applic., 7 (2001), 507-529.
P. Gaborit, Tables of Self-Dual Codes
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105682.

See A105688 for the number of such codes. Cf. also A066012.

A066012 Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105681.

Original entry on oeis.org

2, 2, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 6, 6, 8, 6, 8, 6, 8, 8, 8, 10, 10

Offset: 1

N. J. A. Sloane, Dec 11 2001

S. T. Dougherty, M. Harada and P. Solé, Shadow Codes over Z_4, Finite Fields Applic., 7 (2001), 507-529.
P. Gaborit, Tables of Self-Dual Codes
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.

Cf. A066013 for number of codes. See also A066014-A066017.

A066017 Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 11, 5, 3, 39, 8, 4, 47

Offset: 1

N. J. A. Sloane, Dec 12 2001; revised May 06 2005

There are two versions of this sequence, this and A111259. I am not sure which is correct.

S. T. Dougherty, M. Harada and P. Solé, Shadow Codes over Z_4, Finite Fields Applic., 7 (2001), 507-529.
P. Gaborit, Tables of Self-Dual Codes
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.

Cf. A066015 for minimal distance. See also A066012-A066016.

cp's OEIS Frontend

A066016 Highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.

Views

Author

Keywords

Links

Crossrefs

A105676 Highest minimal Hamming distance of any Type 3 ternary self-dual code of length 4n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A105682 Highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z.

Views

Author

Keywords

Links

Crossrefs

A016729 Highest minimal Hamming distance of any Type 4^H+ Hermitian additive self-dual code over GF(4) of length n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A105675 Highest minimal distance of any Type II doubly-even binary self-dual code of length 8n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A105677 Highest minimal Hamming distance of any Type 4^E Euclidean linear self-dual code over GF(4) of length 2n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A105678 Highest minimal Hamming distance of any Type 4^H Hermitian linear self-dual code over GF(4) of length 2n.

Views

Author

Keywords

Comments

Links

Crossrefs

A105681 Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z.

Views

Author

Keywords

Links

Crossrefs

A066012 Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105681.

Views

Author

Keywords

Links

Crossrefs

A066017 Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^Z self-dual code of length n over Z/4Z.

Views

Author

Keywords

Comments

Links

Crossrefs