Skip to main content
SpringerLink
Log in

New binary quantum stabilizer codes from the binary extremal self-dual \([48, 24, 12]\) code

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

This paper is devoted to constructing binary quantum stabilizer codes based on the binary extremal self-dual code of parameters \([48, 24, 12]\) by Steane’s construction. First, we provide an explicit generator matrix for the unique self-dual \([48,24,12]\) code to see it as a one-generator quasi-cyclic one and obtain six optimal self-orthogonal codes of parameters \([48 - t, 24 - t, 12]\) for \(1 \le t \le 6\) with dual distances from 11 to 7 by puncturing the \([48,24,12]\) code. Second, a special type of subcode structures for self-orthogonal codes is investigated, and then ten derived dual chains are designed. Third, twelve binary quantum codes are constructed from these derived dual pairs within dual chains using Steane’s construction. Ten of them, \([[42,10,8]], [[44,11,8]], [[45,10,8]], [[45,8,9]], [[46,12,8]]\), \([[46,9,9]], [[46,5,10]], [[48,13,8]], [[48,9,9]]\), and \([[48,4,11]]\), achieve as good parameters as the best known ones with comparable lengths and dimensions. Two other codes of parameters \([[47,4,11]]\) and \([[48,3,12]]\) are record breaking in the sense that they improve on the best known ones with the same lengths and dimensions in terms of distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), 2493–2496 (1995)

    Article  ADS  Google Scholar 

  2. Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77(5), 793–797 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996)

    Article  ADS  Google Scholar 

  4. Steane, A.M.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF (4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grassl, M.: Bounds on the minimum distance of additive quantum codes. https://www.iks.kit.edu/home/grassl/codetables/TableIII128.php

  7. Wang, W., Fan, Y., Li, R.: Optimal binary codes and binary construction of quantum codes. Front. Comput. Sci. 8(6), 1024–1031 (2014)

  8. Fan, Y., Wang, W., Li, R.: Binary construction of pure additive quantum codes with distance five or six. Quantum Inf. Process 14(1), 183–200 (2015)

  9. Gulliver, T.A., Bhargava, V.K.: Some best rate \(1/p\) and rate \((p-1)/p\) systematic quasi-cyclic codes. IEEE Trans. Inf. Theory 37(3), 552–555 (1991)

    Article  MathSciNet  Google Scholar 

  10. Daskalov, R., Hristov, P.: New binary one-generator quasi-cyclic codes. IEEE Trans. Inf. Theory 49(11), 3001–3005 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, New York (2003)

    Book  MATH  Google Scholar 

  12. Pless, V.: A classification of self-orthogonal codes over GF (2). Discrete Math. 3(1), 209–246 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mallows, C.L., Sloane, N.J.A.: Weight enumerators of self-orthogonal codes. Discrete Math. 9(4), 391–400 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Houghten, S.K., Lam, C.W.H., Thiel, L.H., Parker, J.A.: The extended quadratic residue code is the only \((48,24,12)\) self-dual doubly-even code. IEEE Trans Inf. Theory 49(1), 53–59 (2003)

    Article  MathSciNet  Google Scholar 

  15. Conan, J., Séguin, G.: Structural properties and enumeration of quasi cyclic codes. Appl. Algebra Eng. Commun. Comput. 4(1), 25–39 (1993)

    Article  MATH  Google Scholar 

  16. Lally, K., Fitzpatrick, P.: Algebraic structure of quasicyclic codes. Discrete Appl. Math. 111(1), 157–175 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cao, Y., Gao, J.: Constructing quasi-cyclic codes from linear algebra theory. Des. Codes Cryptogr. 67(1), 59–75 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the anonymous referees and the editors for their valuable comments and suggestions, which help to improve the manuscript significantly. This work is supported by National Natural Science Foundation of China under Grant No.11471011 and Science Foundation for young teachers in Science College, Air Force Engineering University.

Author information

Authors and Affiliations

  1. School of Electronics and Information, Northwestern Polytechnical University, Xi’an, 710072, China

    WeiLiang Wang & YangYu Fan

  2. Science College, Air Force Engineering University, Xi’an, 710051, China

    WeiLiang Wang & RuiHu Li

Authors
  1. WeiLiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YangYu Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. RuiHu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to WeiLiang Wang.

Appendices

Appendix 1: Basic subcode chains for \(n = 42, 43, 45, 47\)

1.1 \(n = 42\)

$$\begin{aligned}&W_{\mathcal {C}_{42, 18}}(x) =357x^{32} + 12600x^{28} + \cdots + 39522x^{16} + 2744x^{12} + 1.\nonumber \\&W_{\mathcal {C}^{\perp }_{42, 18}}(x) =x^{42} + 48x^{35} + \cdots + 420x^{8} + 48x^{7} + 1. \end{aligned}$$
(7)
$$\begin{aligned}&T^{(42)}_{13 \times 18} = \left( \begin{array}{c|c} 1000000000000 &{} 01010\\ 0100000000000 &{} 00010\\ 0010000000000 &{} 01001\\ 0001000000000 &{} 01111\\ 0000100000000 &{} 10101\\ 0000010000000 &{} 11110\\ 0000001000000 &{} 10001\\ 0000000100000 &{} 01100\\ 0000000010000 &{} 01000\\ 0000000001000 &{} 10000\\ 0000000000100 &{} 00000\\ 0000000000010 &{} 00000\\ 0000000000001 &{} 10000\\ \end{array}\right) . \end{aligned}$$
(8)
$$\begin{aligned}&W_{\mathcal {C}_{42, 13}}(x) =3x^{32} + 420x^{28} + \cdots + 1296x^{16} + 60x^{12} + 1.\nonumber \\&W_{\mathcal {C}^{\perp }_{42, 13}}(x) =x^{42} + 114x^{37} + \cdots + 640x^{6} + 114x^{5} + 1. \end{aligned}$$
(9)

1.2 \(n = 43\)

$$\begin{aligned} W_{\mathcal {C}_{43, 19}}(x)&=8x^{36} + 1365x^{32} + \cdots + 62930x^{16} + 3808x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{43, 19}}(x)&=x^{43} + 8x^{36} + \cdots + 180x^{8} + 8x^{7} + 1. \end{aligned}$$
(10)
$$\begin{aligned} T^{(43)}_{16 \times 19}&= \left( \begin{array}{c|c} 1000000000000000 &{} 101\\ 0100000000000000 &{} 101\\ 0010000000000000 &{} 110\\ 0001000000000000 &{} 111\\ 0000100000000000 &{} 101\\ 0000010000000000 &{} 011\\ 0000001000000000 &{} 000\\ 0000000100000000 &{} 010\\ 0000000010000000 &{} 001\\ 0000000001000000 &{} 001\\ 0000000000100000 &{} 110\\ 0000000000010000 &{} 000\\ 0000000000001000 &{} 001\\ 0000000000000100 &{} 101\\ 0000000000000010 &{} 000\\ 0000000000000001 &{} 000\\ \end{array}\right) , ~ T^{(43)}_{13 \times 16} = \left( \begin{array}{c|c} 1000000000000 &{} 010\\ 0100000000000 &{} 011\\ 0010000000000 &{} 010\\ 0001000000000 &{} 111\\ 0000100000000 &{} 111\\ 0000010000000 &{} 011\\ 0000001000000 &{} 110\\ 0000000100000 &{} 000\\ 0000000010000 &{} 000\\ 0000000001000 &{} 110\\ 0000000000100 &{} 000\\ 0000000000010 &{} 100\\ 0000000000001 &{} 000\\ \end{array}\right) . \end{aligned}$$
(11)
$$\begin{aligned} W_{\mathcal {C}_{43, 16}}(x)&=6x^{36} + 159x^{32} + \cdots + 8020x^{16} + 432x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{43, 16}}(x)&=x^{43} + 118x^{37} + \cdots + 426x^{7} + 118x^{6} + 1.\nonumber \\ W_{\mathcal {C}_{43, 13}}(x)&=14x^{32} + 589x^{28} + \cdots + 1047x^{16} + 33x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{43, 13}}(x)&=x^{43} + 115x^{38} + \cdots + 763x^{6} + 115x^{5} + 1. \end{aligned}$$
(12)

1.3 \(n = 45\)

$$\begin{aligned}&W_{\mathcal {C}_{45, 21}}(x) =220x^{36} + 17325x^{32} + \cdots + 153450x^{16} + 7140x^{12} + 1.\nonumber \\&W_{\mathcal {C}^{\perp }_{45, 21}}(x) =x^{45} + 220x^{36} + \cdots + 2376x^{10} + 220x^{9} + 1. \end{aligned}$$
(13)
$$\begin{aligned}&T^{(45)}_{19 \times 21} = \left( \begin{array}{c|c} 1000000000000000000 &{} 10\\ 0100000000000000000 &{} 10\\ 0010000000000000000 &{} 10\\ 0001000000000000000 &{} 11\\ 0000100000000000000 &{} 00\\ 0000010000000000000 &{} 11\\ 0000001000000000000 &{} 11\\ 0000000100000000000 &{} 10\\ 0000000010000000000 &{} 11\\ 0000000001000000000 &{} 00\\ 0000000000100000000 &{} 11\\ 0000000000010000000 &{} 11\\ 0000000000001000000 &{} 01\\ 0000000000000100000 &{} 11\\ 0000000000000010000 &{} 00\\ 0000000000000001000 &{} 00\\ 0000000000000000100 &{} 10\\ 0000000000000000010 &{} 00\\ 0000000000000000001 &{} 00\\ \end{array}\right) , T^{(45)}_{16 \times 19} = \left( \begin{array}{c|c} 1000000000000000 &{} 011\\ 0100000000000000 &{} 010\\ 0010000000000000 &{} 111\\ 0001000000000000 &{} 011\\ 0000100000000000 &{} 101\\ 0000010000000000 &{} 111\\ 0000001000000000 &{} 100\\ 0000000100000000 &{} 001\\ 0000000010000000 &{} 010\\ 0000000001000000 &{} 110\\ 0000000000100000 &{} 000\\ 0000000000010000 &{} 100\\ 0000000000001000 &{} 000\\ 0000000000000100 &{} 100\\ 0000000000000010 &{} 000\\ 0000000000000001 &{} 000\\ \end{array}\right) .\nonumber \\ \end{aligned}$$
(14)
$$\begin{aligned} W_{\mathcal {C}_{45, 19}}(x)&=34x^{36} + 4473x^{32} + \cdots + 38646x^{16} + 1722x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{45, 19}}(x)&=x^{45} + 108x^{38} + \cdots + 324x^{8} + 108x^{7} + 1.\nonumber \\ W_{\mathcal {C}_{45, 16}}(x)&=7x^{36} + 561x^{32} + 8115x^{28} + \cdots + 4998x^{16} + 169x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{45, 16}}(x)&=x^{45} + 109x^{39} + \cdots + 789x^{7} + 109x^{6} + 1. \end{aligned}$$
(15)
$$\begin{aligned} T^{(45)}_{14 \times 16}&= \left( \begin{array}{c|c} 10000000000000 &{} 01\\ 01000000000000 &{} 00\\ 00100000000000 &{} 01\\ 00010000000000 &{} 10\\ 00001000000000 &{} 11\\ 00000100000000 &{} 11\\ 00000010000000 &{} 01\\ 00000001000000 &{} 10\\ 00000000100000 &{} 00\\ 00000000010000 &{} 10\\ 00000000001000 &{} 00\\ 00000000000100 &{} 10\\ 00000000000010 &{} 00\\ 00000000000001 &{} 00\\ \end{array}\right) . \end{aligned}$$
(16)
$$\begin{aligned} W_{\mathcal {C}_{45, 14}}(x)&=134x^{32} + 2075x^{28} + \cdots + 1255x^{16} + 31x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{45, 14}}(x)&=x^{45} + 65x^{40} + \cdots + 482x^{6} + 65x^{5} + 1. \end{aligned}$$
(17)

1.4 \(n = 47\)

$$\begin{aligned}&W_{\mathcal {C}_{47,23}}(x) = 4324x^{36} + 178365x^{32} + \cdots + 356730x^{16} + 12972x^{12} + 1,\nonumber \\&W_{\mathcal {C}^{\perp }_{47,23}}(x) = x^{47} + 4324x^{36} + \cdots + 12972x^{12} + 4324x^{11} + 1.\end{aligned}$$
(18)
$$\begin{aligned}&T^{(47)}_{20 \times 23} = \left( \begin{array}{c|c} 10000000000000000000 &{} 000\\ 01000000000000000000 &{} 001\\ 00100000000000000000 &{} 111\\ 00010000000000000000 &{} 101\\ 00001000000000000000 &{} 111\\ 00000100000000000000 &{} 011\\ 00000010000000000000 &{} 001\\ 00000001000000000000 &{} 110\\ 00000000100000000000 &{} 011\\ 00000000010000000000 &{} 110\\ 00000000001000000000 &{} 100\\ 00000000000100000000 &{} 010\\ 00000000000010000000 &{} 100\\ 00000000000001000000 &{} 000\\ 00000000000000100000 &{} 000\\ 00000000000000010000 &{} 000\\ 00000000000000001000 &{} 000\\ 00000000000000000100 &{} 000\\ 00000000000000000010 &{} 000\\ 00000000000000000001 &{} 000\\ \end{array}\right) , T^{(47)}_{17 \times 20} = \left( \begin{array}{c|c} 10000000000000000 &{} 000\\ 01000000000000000 &{} 000\\ 00100000000000000 &{} 101\\ 00010000000000000 &{} 010\\ 00001000000000000 &{} 111\\ 00000100000000000 &{} 110\\ 00000010000000000 &{} 101\\ 00000001000000000 &{} 101\\ 00000000100000000 &{} 111\\ 00000000010000000 &{} 110\\ 00000000001000000 &{} 010\\ 00000000000100000 &{} 000\\ 00000000000010000 &{} 100\\ 00000000000001000 &{} 100\\ 00000000000000100 &{} 000\\ 00000000000000010 &{} 000\\ 00000000000000001 &{} 000\\ \end{array}\right) .\nonumber \\ \end{aligned}$$
(19)
$$\begin{aligned} W_{\mathcal {C}_{47, 20}}(x)&=516x^{36} + 22461x^{32} + \cdots + 44922x^{16} + 1548x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{47, 20}}(x)&=x^{47} + 63x^{40} + \cdots + 315x^{8} + 63x^{7} + 1.\nonumber \\ W_{\mathcal {C}_{47, 17}}(x)&=82x^{36} + 2721x^{32} + \cdots + 5694x^{16} + 162x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{47, 17}}(x)&=x^{47} + 84x^{41} + \cdots + 483x^{7} + 84x^{6} + 1. \end{aligned}$$
(20)
$$\begin{aligned} T^{(47)}_{14 \times 17}&= \left( \begin{array}{c|c} 10000000000000 &{} 000\\ 01000000000000 &{} 000\\ 00100000000000 &{} 101\\ 00010000000000 &{} 101\\ 00001000000000 &{} 011\\ 00000100000000 &{} 010\\ 00000010000000 &{} 010\\ 00000001000000 &{} 001\\ 00000000100000 &{} 100\\ 00000000010000 &{} 000\\ 00000000001000 &{} 010\\ 00000000000100 &{} 100\\ 00000000000010 &{} 000\\ 00000000000001 &{} 000\\ \end{array}\right) . \end{aligned}$$
(21)
$$\begin{aligned} W_{\mathcal {C}_{47, 14}}(x)&=5x^{36} + 346x^{32} + \cdots + 707x^{16} + 10x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{47, 14}}(x)&=x^{47} + 88x^{42} + \cdots + 656x^{6} + 88x^{5} + 1. \end{aligned}$$
(22)

Appendix 2: Pairs of nested self-orthogonal codes for \(n = 42,44,46,48\)

1.1 \(n = 42\)

$$\begin{aligned}&W_{\mathcal {C}_{42, 14}}(x) =x^{42} + 3x^{32} + \cdots + 60x^{12} + 3x^{10} + 1.\nonumber \\&W_{\mathcal {C}^{\perp }_{42, 14}}(x) =x^{42} + 640x^{36} + \cdots + 14580x^{8} + 640x^{6} + 1. \end{aligned}$$
(23)
$$\begin{aligned}&W_{\mathcal {C}_{42, 13}}(x) =3x^{32} + 420x^{28} + \cdots + 1296x^{16} + 60x^{12} + 1.\nonumber \\&W_{\mathcal {C}^{\perp }_{42, 13}}(x) =x^{42} + 114x^{37} + \cdots + 640x^{6} + 114x^{5} + 1. \end{aligned}$$
(24)

1.2 \(n = 44\)

$$\begin{aligned}&W_{\mathcal {C}_{44, 14}}(x) =x^{44} + 47x^{32} + \cdots + 1636x^{16} + 47x^{12} + 1.\nonumber \\&W_{\mathcal {C}^{\perp }_{44, 14}}(x) =x^{44} + 878x^{38} + \cdots + 21844x^{8} + 878x^{6} + 1. \end{aligned}$$
(25)
$$\begin{aligned} T^{(44)}_{13 \times 14}&= \left( \begin{array}{c|c} 1000000000000 &{} 1\\ 0100000000000 &{} 0\\ 0010000000000 &{} 1\\ 0001000000000 &{} 0\\ 0000100000000 &{} 1\\ 0000010000000 &{} 1\\ 0000001000000 &{} 0\\ 0000000100000 &{} 0\\ 0000000010000 &{} 0\\ 0000000001000 &{} 0\\ 0000000000100 &{} 0\\ 0000000000010 &{} 0\\ 0000000000001 &{} 0\\ \end{array}\right) . \end{aligned}$$
(26)
$$\begin{aligned} W_{\mathcal {C}_{44, 13}}(x)&=29x^{32} + 818x^{28} + \cdots + 818x^{16} + 18x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{44, 13}}(x)&=x^{44} + 132x^{39} + \cdots + 878x^{6} + 132x^{5} + 1. \end{aligned}$$
(27)

1.3 \(n = 46\)

$$\begin{aligned} W_{\mathcal {C}_{46, 20}}(x)&=x^{46} + 34x^{36} + \cdots + 1722x^{12} + 34x^{10} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{46, 20}}(x)&=x^{46} + 432x^{38} + \cdots + 8404x^{10} + 432x^{8} + 1.\nonumber \\ W_{\mathcal {C}_{46, 19}}(x)&=17x^{36} + 889x^{34} + \cdots + 833x^{12} + 17x^{10} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{46, 19}}(x)&=x^{46} + 116x^{39} + \cdots + 432x^{8} + 116x^{7} + 1. \end{aligned}$$
(28)
$$\begin{aligned} T^{(46)}_{19 \times 20}&= \left( \begin{array}{c|c} 1000000000000000000 &{} 1\\ 0100000000000000000 &{} 1\\ 0010000000000000000 &{} 1\\ 0001000000000000000 &{} 1\\ 0000100000000000000 &{} 0\\ 0000010000000000000 &{} 0\\ 0000001000000000000 &{} 1\\ 0000000100000000000 &{} 0\\ 0000000010000000000 &{} 1\\ 0000000001000000000 &{} 0\\ 0000000000100000000 &{} 0\\ 0000000000010000000 &{} 0\\ 0000000000001000000 &{} 0\\ 0000000000000100000 &{} 0\\ 0000000000000010000 &{} 0\\ 0000000000000001000 &{} 0\\ 0000000000000000100 &{} 0\\ 0000000000000000010 &{} 0\\ 0000000000000000001 &{} 0\\ \end{array}\right) , T^{(46)}_{14 \times 15} = \left( \begin{array}{c|c} 10000000000000 &{} 0\\ 01000000000000 &{} 1\\ 00100000000000 &{} 1\\ 00010000000000 &{} 1\\ 00001000000000 &{} 1\\ 00000100000000 &{} 0\\ 00000010000000 &{} 0\\ 00000001000000 &{} 0\\ 00000000100000 &{} 0\\ 00000000010000 &{} 0\\ 00000000001000 &{} 0\\ 00000000000100 &{} 0\\ 00000000000010 &{} 0\\ 00000000000001 &{} 0\\ \end{array}\right) . \end{aligned}$$
(29)
$$\begin{aligned} W_{\mathcal {C}_{46, 15}}(x)&=x^{46} + 31x^{34} + \cdots + 134x^{14} + 31x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{46, 15}}(x)&=x^{46} + 547x^{40} + \cdots + 16127x^{8} + 547x^{6} + 1.\nonumber \\ W_{\mathcal {C}_{46, 14}}(x)&=19x^{34} + 65x^{32} + \cdots + 69x^{14} + 12x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{46, 14}}(x)&=x^{46} + 86x^{41} + \cdots + 547x^{6} + 86x^{5} + 1. \end{aligned}$$
(30)

1.4 \(n = 48\)

$$\begin{aligned} W_{\mathcal {C}_{48, 21}}(x)&=x^{48} + 2064x^{36} + \cdots + 67383x^{16} + 2064x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{48, 21}}(x)&=x^{48} + 378x^{40} + \cdots + 5376x^{10} + 378x^{8} + 1.\nonumber \\ W_{\mathcal {C}_{48, 20}}(x)&=1060x^{36} + 33597x^{32} + \cdots + 33786x^{16} + 1004x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{48, 20}}(x)&=x^{48} + 72x^{41} + \cdots + 378x^{8} + 72x^{7} + 1. \end{aligned}$$
(31)
$$\begin{aligned} T^{(48)}_{20 \times 21}&= \left( \begin{array}{c|c} 10000000000000000000 &{} 1\\ 01000000000000000000 &{} 1\\ 00100000000000000000 &{} 1\\ 00010000000000000000 &{} 1\\ 00001000000000000000 &{} 1\\ 00000100000000000000 &{} 0\\ 00000010000000000000 &{} 0\\ 00000001000000000000 &{} 1\\ 00000000100000000000 &{} 0\\ 00000000010000000000 &{} 0\\ 00000000001000000000 &{} 0\\ 00000000000100000000 &{} 0\\ 00000000000010000000 &{} 0\\ 00000000000001000000 &{} 0\\ 00000000000000100000 &{} 0\\ 00000000000000010000 &{} 0\\ 00000000000000001000 &{} 0\\ 00000000000000000100 &{} 0\\ 00000000000000000010 &{} 0\\ 00000000000000000001 &{} 0\\ \end{array}\right) , T^{(48)}_{14 \times 15} = \left( \begin{array}{c|c} 10000000000000 &{} 1\\ 01000000000000 &{} 1\\ 00100000000000 &{} 1\\ 00010000000000 &{} 1\\ 00001000000000 &{} 0\\ 00000100000000 &{} 0\\ 00000010000000 &{} 0\\ 00000001000000 &{} 0\\ 00000000100000 &{} 0\\ 00000000010000 &{} 0\\ 00000000001000 &{} 0\\ 00000000000100 &{} 0\\ 00000000000010 &{} 0\\ 00000000000001 &{} 0\\ \end{array}\right) . \end{aligned}$$
(32)
$$\begin{aligned} W_{\mathcal {C}_{48, 15}}(x)&=x^{48} + 15x^{36} + \cdots + 1053x^{16} + 15x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{48, 15}}(x)&=x^{48} + 744x^{42} + \cdots + 23130x^{8} + 744x^{6} + 1.\nonumber \\ W_{\mathcal {C}_{48, 14}}(x)&=9x^{36} + 538x^{32} + \cdots + 515x^{16} + 6x^{12} + 1.\nonumber \\ W_{\mathcal {C}^{\perp }_{48, 14}}(x)&=x^{48} + 106x^{43} + \cdots + 744x^{6} + 106x^{5} + 1. \end{aligned}$$
(33)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, W., Fan, Y. & Li, R. New binary quantum stabilizer codes from the binary extremal self-dual \([48, 24, 12]\) code. Quantum Inf Process 14, 2761–2774 (2015). https://doi.org/10.1007/s11128-015-1018-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-1018-9

Keywords

Search

Navigation