A066014
Highest minimal Euclidean norm 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 A105682.
Original entry on oeis.org
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 12, 12
Offset: 1
- 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).
A105685
Number of inequivalent codes attaining highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 1, 2, 7, 1, 1, 1, 3, 13, 3
Offset: 1
At length 8 the only strictly Type I self-dual code is {00,11}^4, so a(4) = 1.
- J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53.
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.
- V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746.
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A28 (1980), 26-53 (Abstract, pdf, ps, Table A, Table D).
- 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).
A105674 gives the minimal distance of these codes,
A106165 the number of codes of any minimal distance and
A003179 the number of inequivalent codes allowing Type I or Type II and any minimal distance.
A066015
Number of codes having highest minimal Euclidean norm 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 A105682.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 6, 11, 16, 19, 19, 66, 35, 28
Offset: 1
- 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).
A105687
Number of inequivalent codes attaining 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, 1, 1, 3, 1, 1, 4, 5, 8, 120, 1, 1
Offset: 1
- C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, in International Workshop on Coding and Cryptography (Paris, 2001), Electron. Notes Discrete Math. 6 (2001), 10 pp.
- P. Gaborit, W. C. Huffman, J.-L. Kim and V. S. Pless, On additive GF(4) codes, in Codes and Association Schemes (Piscataway, NJ, 1999), A. Barg and S. Litsyn, eds., Amer. Math. Soc., Providence, RI, 2001, pp. 135-149.
- G. Hoehn, Self-dual codes over the Kleinian four-group, Math. Ann. 327 (2003), 227-255.
- 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), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
- L. E. Danielsen, Database of Self-Dual Quantum Codes.
- L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, preprint, 2005.
- 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.
A094927,
A090899,
A105674,
A105675,
A105676,
A105677,
A105678,
A016729,
A066016,
A105681,
A105682.
A016729 gives the minimal distance of these codes.
A094927 gives the number of inequivalent codes of any distance.
Corrected and extended to 12 terms by Lars Eirik Danielsen (larsed(AT)ii.uib.no) and Matthew G. Parker (matthew(AT)ii.uib.no), Jun 30 2005
A105688
Number of codes having highest minimal Lee 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, 1, 15
Offset: 1
- 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).
A105689
Number of codes having highest minimal Euclidean norm of any Type 4^Z self-dual code of length n over Z/4Z.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 4, 1, 11, 16, 19, 19, 66, 35, 28
Offset: 1
- 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).
A066013
Number of codes having 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 A105688.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 11, 5, 3, 39, 8, 1, 15
Offset: 1
- 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).
A105686
Number of inequivalent codes attaining 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
1, 1, 1, 1, 2, 5, 1, 4, 1, 2
Offset: 1
- P. Gaborit, Tables of Self-Dual Codes
- W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28. I, IEEE Trans. Infor. Theory, 36 (1990), 651-660.
- W. C. Huffman, On extremal self-dual quaternary codes of lengths 18 to 28. II, IEEE Trans. Infor. Theory, 37 (1991), 1206-1216.
- W. C. Huffman, On 3-elements in monomial automorphism groups of quaternary codes, IEEE Trans. Infor. Theory, 36 (1990), 660-664.
- 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).
A105678 gives the minimal distance of these codes.
A106169
Number of inequivalent codes attaining highest minimal Hamming distance of any Type (4_II)^H+ even Hermitian additive self-dual code over GF(4) of length 2n.
Original entry on oeis.org
1, 2, 1, 3, 19, 1, 1020
Offset: 1
- C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, in International Workshop on Coding and Cryptography (Paris, 2001), Electron. Notes Discrete Math. 6 (2001), 10 pp.
- 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, W. C. Huffman, J.-L. Kim and V. S. Pless, On additive GF(4) codes, in Codes and Association Schemes (Piscataway, NJ, 1999), A. Barg and S. Litsyn, eds., Amer. Math. Soc., Providence, RI, 2001, pp. 135-149.
- G. Hoehn, Self-dual codes over the Kleinian four-group, Math. Ann. 327 (2003), 227-255.
- W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
- W. C. Huffman, Additive self-dual codes over F_4 with an automorphism of odd prime order, Adv. Math. Commun., 1 (2007), 357-398.
- 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).
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