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## self-dual codes of length 38 (1111)

### Citations

76 | A new upper bound on the minimal distance of self-dual codes
- Conway, Sloane
- 1990
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Citation Context ...[38,19,8] SELF-DUAL CODES A. Construction of all [38,19,8] self-dual codes There are two possible weight enumerators W1,W2 and shadow weight enumerators S1,S2 for an extremal self-dual [38,19,8] code =-=[9]-=-. W1 = 1+171y 8 +1862y 10 +··· (4) S1 = 114y 7 +9044y 11 +118446y 15 +··· ; (5) W2 = 1+203y 8 +1702y 10 +··· (6) S2 = y 3 +106y 7 +9072y 11 +118390y 15 In [9] two self-dual [38,19,8] codes with W1, de... |

22 |
On the classification and enumeration of self-dual codes
- Huffman
- 2005
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Citation Context ...w of a code to the that of the shadow ⎠ of the subtracted code under certain conditions. We use this result to prove the following classification theorem: Theorem 10: There are exactly two s-extremal =-=[38,19,6]-=- codes. Proof. Let C be an s-extremal [38,19,6] code, then C has shadow weight s = 11. Applying Theorem 8 we deduce that there exist two coordinates on which the subtraction of (11) of C produces a [3... |

19 | Extremal binary self-dual codes - Dougherty, Gulliver, et al. - 1997 |

12 | Designs and self-dual codes with long shadows
- Bachoc, Gaborit
- 2004
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Citation Context ... we classify s-extremal codes of length 38 and d = 8 together with s-extremal codes of length 38 and d = 6 . A. s-extremal codes The notion of s-extremal codes was introduced by Bachoc and Gaborit in =-=[2]-=-. This type of codes is related to the notion of self-dual codes with long shadows introduced by Elkies in [11]. We recall the definition of s-extremal codes from [2]. Let C be a Type I self-dual bina... |

8 |
A complete classification of doubly even self-dual codes of length 40
- Betsumiya, Harada, et al.
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Citation Context ...□ ACKNOWLEDGEMENT All the computations were done with the MAGMA system [4]. While this work was under review, the authors learned that independently, similar results have been obtained by others (see =-=[3]-=-, [5]). APPENDIX Let i = 6,9,12,14,18,21,24,36,144,168,216,342,504. Then G(Ci 38) represents a generator matrix of a new selfdual [38,19,8] code Ci 38 with the automorphism group order |Aut(Ci 38 )| =... |

8 |
New extremal self-dual codes of lengths 36 and 38
- Harada
- 1999
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Citation Context ...lf-dual [38,19,8] codes with W1, denoted by R3 and D4, were given, where |Aut(R3)| = 1 and |Aut(D4)| = 342. In [16] one self-dual [38,19,8] code C38 with W2 was given with |Aut(C38)| = 1. Then Harada =-=[15]-=- gave 40 self-dual [38,19,8] codes with W1 and W2 and automorphism group orders 1,2,4,8. Later, Kim [21] constructed 325 self-dual [38,19,8] codes with W1 and W2 and (7) automorphism group orders1,2,3... |

7 |
Elkies Lattices and codes with long shadows
- D
- 1995
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Citation Context ...ual codes whose weight enumerator is uniquely determined, depending on the condition on a high weight of the shadow. The notion of codes (and lattices) with long shadows was first developed by Elkies =-=[11]-=-. This notion was generalized by Bachoc and Gaborit in [2] who introduced the notion of s-extremal codes. These codes exist depending on conditions on their length and their minimum distance. The clas... |

6 |
Singly-even self-dual codes of length 40, Des
- Buyuklieva, Yorgov
- 1996
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Citation Context ...odes of length 38 and d = 8 from Theorem 9. We list currently known codes for d = 8 in Table V. TABLE V NUMBER OF s-EXTREMAL CODES WITHd = 8 n num ref 32 3 [9] 36 25 [1] 38 1730 this paper 40 ≥ 4 [9],=-=[7]-=- 42 ≥ 17 [9],[6] 44 ≥ 1 [9] VI. COVERING RADII OF SELF-DUAL CODES OF LENGTH 38 The covering radiusρ(C) of a codeC is the smallest integer R such that spheres of radius R around codewords cover F n 2 .... |

6 |
Database of self-dual codes
- Harada, Munemasa
- 2007
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Citation Context ...(5) W2 = 1+203y 8 +1702y 10 +··· (6) S2 = y 3 +106y 7 +9072y 11 +118390y 15 In [9] two self-dual [38,19,8] codes with W1, denoted by R3 and D4, were given, where |Aut(R3)| = 1 and |Aut(D4)| = 342. In =-=[16]-=- one self-dual [38,19,8] code C38 with W2 was given with |Aut(C38)| = 1. Then Harada [15] gave 40 self-dual [38,19,8] codes with W1 and W2 and automorphism group orders 1,2,4,8. Later, Kim [21] constr... |

6 |
Characterization of quaternary extremal codes of lengths 18 and 20
- Huffman
- 1997
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Citation Context ...ively small number of possibilities of a ′ i s in Step 2), whose complexity is 2n/2−k , where k ≥ 1 depends on the given code. From our experimental results, the dimensions k of subcodes of the 58671 =-=[36,18,6]-=- codes generated by linearly independent vectors of weight 6 lie between 2 and 18. We give the possible values of k and the number num of their subcodes in Table I. TABLE I NUMBER OF SELF-DUAL[36,18,6... |

5 |
A bound for certain s-extremal lattices and codes
- Gaborit
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Citation Context ...the previous bounds is called s-extremal. In that case, the polynomials WC and WS are uniquely determined. A bound fornwhen the minimum weightdof ans-extremal code is divisible by 4 has been given in =-=[12]-=- and in [14], and a bound has also been given for d = 6 [2, Theorem 4.1] and d ≡ 2 (mod 4) with d > 6 [14]. Theorem 6: ([12], [14]) LetC be ans-extremal code with parameters(s,d) of length n. If d ≡ 0... |

4 |
On the classification of extremal
- Melchor, Gaborit
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Citation Context ... recursive method, Aguilar and Gaborit classified all 41 extremal [36,18,8] binary self-dual codes. These results were pushed further by Harada and Munemasa [17] who, besides the 41 extremal codes of =-=[1]-=-, also give a complete classification of all self-dual codes of length 36. A natural question is hence to consider the case of length 38. A simple computation on the mass formula shows that there are ... |

4 |
Covering radius--survey and recent results
- Schatz
- 1985
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Citation Context ... Kim, Lin Sok and Patrick Solé arXiv:1111.0228v1 [cs.DM] 1 Nov 2011 Abstract—In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal =-=[38,19,8]-=- self-dual codes, two s-extremal [38,19,6] codes, and 1730 s-extremal [38,19,8] codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal sel... |

4 |
Upper bounds for the length of s-extremal codes over
- Han, Kim
- 2007
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Citation Context ... bounds is called s-extremal. In that case, the polynomials WC and WS are uniquely determined. A bound fornwhen the minimum weightdof ans-extremal code is divisible by 4 has been given in [12] and in =-=[14]-=-, and a bound has also been given for d = 6 [2, Theorem 4.1] and d ≡ 2 (mod 4) with d > 6 [14]. Theorem 6: ([12], [14]) LetC be ans-extremal code with parameters(s,d) of length n. If d ≡ 0 (mod 4), th... |

3 |
New extremal self-dual codes of length 42 and 44
- Buyuklieva
- 1997
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Citation Context ...0228v1 [cs.DM] 1 Nov 2011 Abstract—In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal [38,19,8] self-dual codes, two s-extremal =-=[38,19,6]-=- codes, and 1730 s-extremal [38,19,8] codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a gene... |

2 |
On the classification of binary self-dual codes, arXiv:1106.5930v1
- Bouyuklieva, Bouyukliev
- 2011
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Citation Context ...NOWLEDGEMENT All the computations were done with the MAGMA system [4]. While this work was under review, the authors learned that independently, similar results have been obtained by others (see [3], =-=[5]-=-). APPENDIX Let i = 6,9,12,14,18,21,24,36,144,168,216,342,504. Then G(Ci 38) represents a generator matrix of a new selfdual [38,19,8] code Ci 38 with the automorphism group order |Aut(Ci 38 )| = i. ⎡... |

2 |
Construction of some extremal self-dual codes
- Gulliver, Harada, et al.
- 2003
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Citation Context ...g-up construction [21] needs 2 n−1 possibilities for the choice of odd vectors x, generating all self-dual codes with various minimum distances. This complexity can be reduced to 2 n/2 as remarked in =-=[13]-=-, which is still higher than that of the recursive construction. As described above, Harada-Munemasa’s construction is effective if the given code has a large automorphism group in order to reduce the... |

1 |
Classification of self-dual codes of length 36,” arXiv:1012.5464v1 [math.CO
- Harada, Munemasa
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Citation Context ...es over small alphabets). Recently, using a recursive method, Aguilar and Gaborit classified all 41 extremal [36,18,8] binary self-dual codes. These results were pushed further by Harada and Munemasa =-=[17]-=- who, besides the 41 extremal codes of [1], also give a complete classification of all self-dual codes of length 36. A natural question is hence to consider the case of length 38. A simple computation... |