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Journal articles on the topic 'Extremal self-dual codes'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

Dougherty, S. T., T. A. Gulliver, and M. Harada. "Extremal binary self-dual codes." IEEE Transactions on Information Theory 43, no. 6 (1997): 2036–47. http://dx.doi.org/10.1109/18.641574.

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2

Dontcheva, R., and M. Harada. "New extremal self-dual codes of length 62 and related extremal self-dual codes." IEEE Transactions on Information Theory 48, no. 7 (2002): 2060–64. http://dx.doi.org/10.1109/tit.2002.1013144.

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3

Bouyuklieva, Stefka, Anton Malevich, and Wolfgang Willems. "Automorphisms of Extremal Self-Dual Codes." IEEE Transactions on Information Theory 56, no. 5 (2010): 2091–96. http://dx.doi.org/10.1109/tit.2010.2043763.

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4

Han, Sunghyu. "Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4)." Information 9, no. 7 (2018): 172. http://dx.doi.org/10.3390/info9070172.

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Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22.

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5

Karadeniz, Suat, Bahattin Yildiz, and Nuh Aydin. "Extremal binary self-dual codes of lengths 64 and 66 from four-circulant constructions over F2+uF2." Filomat 28, no. 5 (2014): 937–45. http://dx.doi.org/10.2298/fil1405937k.

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A classification of all four-circulant extremal codes of length 32 over F2 + uF2 is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with ?=80 in W64,2 is found. Then applying an extension method from the literature to extremal self-dual codes of length 64, we have found many extremal binary self-dual codes of length 66. Among those, five of them are new codes in the sense that codes with these weight enumerators

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6

Pender, Thomas. "On extremal and near-extremal self-dual ternary codes." Discrete Mathematics 347, no. 6 (2024): 113968. http://dx.doi.org/10.1016/j.disc.2024.113968.

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7

GABORIT, PHILIPPE, ANN MARIE NATIVIDAD, and PATRICK SOLÉ. "EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES." International Journal of Number Theory 02, no. 02 (2006): 289–303. http://dx.doi.org/10.1142/s1793042106000577.

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Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension

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8

Han, Sun-Ghyu, and June-Bok Lee. "NONEXISTENCE OF SOME EXTREMAL SELF-DUAL CODES." Journal of the Korean Mathematical Society 43, no. 6 (2006): 1357–69. http://dx.doi.org/10.4134/jkms.2006.43.6.1357.

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9

Tsai, H. P. "Existence of certain extremal self-dual codes." IEEE Transactions on Information Theory 38, no. 2 (1992): 501–4. http://dx.doi.org/10.1109/18.119711.

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10

Han-Ping Tsai. "Existence of some extremal self-dual codes." IEEE Transactions on Information Theory 38, no. 6 (1992): 1829–33. http://dx.doi.org/10.1109/18.165461.

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11

Han-Ping Tsai and Yih-Jaw Jiang. "Some new extremal self-dual [58,29,10] codes." IEEE Transactions on Information Theory 44, no. 2 (1998): 813–14. http://dx.doi.org/10.1109/18.661527.

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12

Gulliver, T. Aaron, Masaaki Harada, and Jon-Lark Kim. "Construction of new extremal self-dual codes." Discrete Mathematics 263, no. 1-3 (2003): 81–91. http://dx.doi.org/10.1016/s0012-365x(02)00570-8.

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13

Koch, Helmut. "On self-dual doubly-even extremal codes." Discrete Mathematics 83, no. 2-3 (1990): 291–300. http://dx.doi.org/10.1016/0012-365x(90)90013-8.

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14

Spence, Edward, and Vladimir D. Tonchev. "Extremal self-dual codes from symmetric designs." Discrete Mathematics 110, no. 1-3 (1992): 265–68. http://dx.doi.org/10.1016/0012-365x(92)90716-s.

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15

Guan, Chaofeng, Ruihu Li, Hao Song, Liangdong Lu, and Husheng Li. "Ternary quantum codes constructed from extremal self-dual codes and self-orthogonal codes." AIMS Mathematics 7, no. 4 (2022): 6516–34. http://dx.doi.org/10.3934/math.2022363.

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<abstract><p>In this paper, we consider $ S $-chains of extremal self-dual and self-orthogonal codes and their applications in the construction of quantum codes. Then, by virtue of covering radius, we determine necessary conditions for linear codes to have subcodes with large dual distances and design a new $ S $-chain search method. As computational results, 18 $ S $-chains with large distances are obtained, and many good quantum codes can be derived from those $ S $-chains by Steane construction, some of which improve the previous results.</p></abstract>

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16

Borello, Martino, and Gabriele Nebe. "On involutions in extremal self-dual codes and the dual distance of semi self-dual codes." Finite Fields and Their Applications 33 (May 2015): 80–89. http://dx.doi.org/10.1016/j.ffa.2014.11.008.

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17

Varbanov, Zlatko. "Some new Results for Additive Self-Dual Codes over GF(4)." Serdica Journal of Computing 1, no. 2 (2007): 213–27. http://dx.doi.org/10.55630/sjc.2007.1.213-227.

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Additive code C over GF(4) of length n is an additive subgroup of GF(4)n. It is well known [4] that the problem of finding stabilizer quantum error-correcting codes is transformed into problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to construct good additive self-dual codes of length 13 ≤ n ≤ 21. In this paper we classify all extremal (optimal) codes of lengths 13 and 14, and we construct many extremal codes of lengths 15 and 16. Also, we construct some new extremal codes of lengths 17,18,19, and 21. We give the current

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18

Mravić, Matteo, and Sanja Rukavina. "A Note on New Near-Extremal Type I Z4-Codes of Length 48." Mathematics 13, no. 6 (2025): 946. https://doi.org/10.3390/math13060946.

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The subject of this work is self-dual Type I Z4-codes of length 48. It is known that the usual upper bound for the minimum Euclidean weight for the self-dual Z4-codes cannot be achieved for this length and type of codes; i.e., extremal Type I Z4-codes of length 48 do not exist. We are therefore looking for near-extremal Type I Z4-codes of length 48. The only known Type I near-extremal Z4-code of length 48 was constructed in 2015 by M. Harada. We adapted a known pseudo-random search method and found at least two new near-extremal Type I Z4-codes of length 48. These codes are the first Type I ne

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19

BETSUMIYA, Koichi, T. Aaron GULLIVER, and Masaaki HARADA. "ON BINARY EXTREMAL FORMALLY SELF-DUAL EVEN CODES." Kyushu Journal of Mathematics 53, no. 2 (1999): 421–30. http://dx.doi.org/10.2206/kyushujm.53.421.

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20

Dougherty, S. T., and M. Harada. "New extremal self-dual codes of length 68." IEEE Transactions on Information Theory 45, no. 6 (1999): 2133–36. http://dx.doi.org/10.1109/18.782158.

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21

de la Cruz, Javier, and Wolfgang Willems. "On Extremal Self-Dual Codes of Length 96." IEEE Transactions on Information Theory 57, no. 10 (2011): 6820–23. http://dx.doi.org/10.1109/tit.2011.2155031.

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22

Shengyuan, Zhang. "On the nonexistence of extremal self-dual codes." Discrete Applied Mathematics 91, no. 1-3 (1999): 277–86. http://dx.doi.org/10.1016/s0166-218x(98)00131-0.

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23

de la Cruz, Javier. "On extremal self-dual codes of length 120." Designs, Codes and Cryptography 75, no. 2 (2013): 243–52. http://dx.doi.org/10.1007/s10623-013-9902-8.

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24

Nebe, Gabriele. "On Extremal Self-Dual Ternary Codes of Length 48." International Journal of Combinatorics 2012 (January 23, 2012): 1–9. http://dx.doi.org/10.1155/2012/154281.

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25

Harada, Masaaki. "Binary extremal self-dual codes of length 60 and related codes." Designs, Codes and Cryptography 86, no. 5 (2017): 1085–94. http://dx.doi.org/10.1007/s10623-017-0380-2.

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26

Dougherty, Steven T., Joseph Gildea, Adrian Korban, Abidin Kaya, Alexander Tylyshchak, and Bahattin Yildiz. "Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes." Finite Fields and Their Applications 57 (May 2019): 108–27. http://dx.doi.org/10.1016/j.ffa.2019.02.004.

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27

Bouyuklieva, Stefka, Anton Malevich, and Wolfgang Willems. "On the performance of binary extremal self-dual codes." Advances in Mathematics of Communications 5, no. 2 (2011): 267–74. http://dx.doi.org/10.3934/amc.2011.5.267.

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28

Georgiou, S., C. Koukouvinos, and E. Lappas. "Extremal doubly-even self-dual codes from Hadamard matrices." Journal of Discrete Mathematical Sciences and Cryptography 9, no. 2 (2006): 331–39. http://dx.doi.org/10.1080/09720529.2006.10698082.

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29

Tonchev, V., and V. Y. Yorgov. "The existence of certain extremal [54,27,10] self-dual codes." IEEE Transactions on Information Theory 42, no. 5 (1996): 1628–31. http://dx.doi.org/10.1109/18.532913.

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30

Han-Ping Tsai. "Extremal self-dual codes of lengths 66 and 68." IEEE Transactions on Information Theory 45, no. 6 (1999): 2129–33. http://dx.doi.org/10.1109/18.782156.

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31

Kaya, Abidin, and Bahattin Yildiz. "New extremal binary self-dual codes of length 68." Journal of Algebra Combinatorics Discrete Structures and Applications 1, no. 1 (2014): 29. http://dx.doi.org/10.13069/jacodesmath.79879.

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32

Harada, Masaaki, W. Holzmann, H. Kharaghani, and M. Khorvash. "Extremal Ternary Self-Dual Codes Constructed from Negacirculant Matrices." Graphs and Combinatorics 23, no. 4 (2007): 401–17. http://dx.doi.org/10.1007/s00373-007-0731-2.

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33

Harada, Masaaki, Akihiro Munemasa, and Kenichiro Tanabe. "Extremal self-dual [40,20,8] codes with covering radius 7." Finite Fields and Their Applications 10, no. 2 (2004): 183–97. http://dx.doi.org/10.1016/j.ffa.2003.08.001.

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34

Kim, Hyun Jin, and Yoonjin Lee. "Extremal quasi-cyclic self-dual codes over finite fields." Finite Fields and Their Applications 52 (July 2018): 301–18. http://dx.doi.org/10.1016/j.ffa.2018.04.013.

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35

Han, Sunghyu, and Jon-Lark Kim. "The nonexistence of near-extremal formally self-dual codes." Designs, Codes and Cryptography 51, no. 1 (2008): 69–77. http://dx.doi.org/10.1007/s10623-008-9244-0.

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36

Harada, Masaaki, and Michio Ozeki. "Extremal self-dual codes with the smallest covering radius." Discrete Mathematics 215, no. 1-3 (2000): 271–81. http://dx.doi.org/10.1016/s0012-365x(99)00318-0.

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37

Karadeniz, Suat, and Bahattin Yildiz. "New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes." Advances in Mathematics of Communications 7, no. 2 (2013): 219–29. http://dx.doi.org/10.3934/amc.2013.7.219.

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38

Karadeniz, Suat, and Bahattin Yildiz. "New extremal binary self-dual codes of length 66 as extensions of self-dual codes over Rk." Journal of the Franklin Institute 350, no. 8 (2013): 1963–73. http://dx.doi.org/10.1016/j.jfranklin.2013.05.015.

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39

Yorgova, Radinka, and Nuray At. "On Extremal Binary Doubly-Even Self-Dual Codes of Length 88*." Serdica Journal of Computing 3, no. 3 (2009): 239–48. http://dx.doi.org/10.55630/sjc.2009.3.239-248.

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In this paper we present 35 new extremal binary self-dual doubly-even codes of length 88. Their inequivalence is established by invariants. Moreover, a construction of a binary self-dual [88, 44, 16] code, having an automorphism of order 21, is given.

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40

Aguilar-Melchor, Carlos, Philippe Gaborit, Jon-Lark Kim, Lin Sok, and Patrick Sole. "Classification of Extremal and $s$-Extremal Binary Self-Dual Codes of Length 38." IEEE Transactions on Information Theory 58, no. 4 (2012): 2253–62. http://dx.doi.org/10.1109/tit.2011.2177809.

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41

Harada, Masaaki, and Katsushi Waki. "New extremal formally self-dual even codes of length 30." Advances in Mathematics of Communications 3, no. 4 (2009): 311–16. http://dx.doi.org/10.3934/amc.2009.3.311.

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42

Harada, Masaaki. "Self-dual codes over F5 and s-extremal unimodular lattices." Discrete Mathematics 346, no. 1 (2023): 113126. http://dx.doi.org/10.1016/j.disc.2022.113126.

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43

Buyuklieva, S. "New extremal self-dual codes of lengths 42 and 44." IEEE Transactions on Information Theory 43, no. 5 (1997): 1607–12. http://dx.doi.org/10.1109/18.623159.

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44

Buyuklieva, S., and N. Boukliev. "Extremal self-dual codes with an automorphism of order 2." IEEE Transactions on Information Theory 44, no. 1 (1998): 323–28. http://dx.doi.org/10.1109/18.651059.

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45

Harada, M. "New extremal self-dual codes of lengths 36 and 38." IEEE Transactions on Information Theory 45, no. 7 (1999): 2541–43. http://dx.doi.org/10.1109/18.796402.

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46

Harada, Masaaki, Michio Ozeki, and Kenichiro Tanabe. "On the Covering Radius of Ternary Extremal Self-Dual Codes." Designs, Codes and Cryptography 33, no. 2 (2004): 149–58. http://dx.doi.org/10.1023/b:desi.0000035468.86695.40.

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47

Harada, Masaaki, Akihiro Munemasa, and Boris Venkov. "Classification of ternary extremal self-dual codes of length 28." Mathematics of Computation 78, no. 267 (2009): 1787–96. http://dx.doi.org/10.1090/s0025-5718-08-02194-7.

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48

Aguilar Melchor, Carlos, and Philippe Gaborit. "On the Classification of Extremal $[36,18,8]$ Binary Self-Dual Codes." IEEE Transactions on Information Theory 54, no. 10 (2008): 4743–50. http://dx.doi.org/10.1109/tit.2008.928976.

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49

T. Dougherty, Steven, Joe Gildea, Adrian Korban, and Abidin Kaya. "Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68." Advances in Mathematics of Communications 14, no. 4 (2020): 677–702. http://dx.doi.org/10.3934/amc.2020037.

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50

Harada, Masaaki. "Extremal Type IIZ4-codes constructed from binary doubly even self-dual codes of length40." Discrete Mathematics 340, no. 10 (2017): 2466–68. http://dx.doi.org/10.1016/j.disc.2017.06.009.

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