Abstract

The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field ${\rm GF}(q)$ is constructed by assigning nonzero elements from ${\rm GF}(q)$ to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over ${\rm GF}(q),$ constructed by assigning nonzero elements from ${\rm GF}(q)$ to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of $10^{-12}$. Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.

© 2009 IEEE

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  1. R. G. Gallager, "Low density parity check codes," IRE Trans. Inf. Theory IT-8, 21-28 (1962).
  2. D. J. C. MacKay, R. M. Neal, "Near Shannon limit performance of low density parity check codes," Electron. Lett. 33, 457-458 (1997).
  3. D. J. C. MacKay, "Good error-correcting codes based on very sparse matrices," IEEE Trans. Inf. Theory 45, 399-431 (1999).
  4. T. Ohtsuki, "LDPC codes in communications and broadcasting," IEICE Trans. Commun. E90-B, 440-453 (2007).
  5. A. Dholakia, E. Eleftheriou, T. Mittelholzer, M. P. C. Fossorier, "Capacity-approaching codes: Can they be applied to the magnetic recording channel?," IEEE Commun. Mag. 42, 122-130 (2004).
  6. M. C. Davey, Error-correction using low-density parity-check codes Ph.D. dissertation Univ. CambridgeCambridgeU.K. (1999).
  7. D. Declercq, M. Fossorier, "Decoding algorithms for nonbinary LDPC codes over ${\rm GF}(q)$," IEEE Trans. Commun. 55, 633-643 (2007).
  8. C. Spagnol, W. Marnane, E. Popovici, "FPGA implementations of LDPC over ${\rm GF}(2^{m})$ decoders," Proc. IEEE Workshop Signal Process. Syst. (2007) pp. 273-278.
  9. Z.-W. Li, L. Chen, L.-Q. Zeng, S. Lin, W. Fong, "Efficient encoding of quasi-cyclic low-density parity-check codes," IEEE Trans. Commun. 54, 71-81 (2006).
  10. S. Lin, D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications (Prentice-Hall, 2004).
  11. Z. Wang, Z. Cui, "Low-complexity high-speed decoder design for quasi-cyclic LDPC codes," IEEE Trans. Very Large Scale Integr. Syst. 15, 104-114 (2007).
  12. Y. Chen, K. K. Parhi, "Overlapped message passing for quasi-cyclic low-density parity check codes," IEEE Trans. Circuits Syst. I, Reg. Papers 51, 1106-1113 (2004) .
  13. L. Lan, L. Zeng, Y. Y. Tai, L. Chen, S. Lin, K. Abdel-Ghaffar, "Construction of quasi-cyclic LDPC codes for AWGN and binary erasure channels: A finite field approach," IEEE Trans. Inf. Theory 53, 2429-2458 (2007).
  14. R. Tanner, D. Sridhara, A. Sridharan, T. Fuja, D. Costello, Jr., "LDPC block and convolutional codes based on circulant matricies," IEEE Trans. Inf. Theory 50, 2966-2984 (2004).
  15. M. Arabaci, I. B. Djordjevic, R. Saunders, R. Marcoccia, "A class of non-binary regular girth-8 LDPC codes for optical communication channels," Proc. OFC/NFOEC 2009 ().
  16. B. Rong, T. Jiang, X. Li, M. R. Soleymani, "Combine LDPC codes over ${\rm GF}(q)$ with $q$-ary modulations for bandwidth-efficient transmission," IEEE Trans. Broadcast. 54, 78-84 (2008).

2008 (1)

B. Rong, T. Jiang, X. Li, M. R. Soleymani, "Combine LDPC codes over ${\rm GF}(q)$ with $q$-ary modulations for bandwidth-efficient transmission," IEEE Trans. Broadcast. 54, 78-84 (2008).

2007 (4)

L. Lan, L. Zeng, Y. Y. Tai, L. Chen, S. Lin, K. Abdel-Ghaffar, "Construction of quasi-cyclic LDPC codes for AWGN and binary erasure channels: A finite field approach," IEEE Trans. Inf. Theory 53, 2429-2458 (2007).

T. Ohtsuki, "LDPC codes in communications and broadcasting," IEICE Trans. Commun. E90-B, 440-453 (2007).

D. Declercq, M. Fossorier, "Decoding algorithms for nonbinary LDPC codes over ${\rm GF}(q)$," IEEE Trans. Commun. 55, 633-643 (2007).

Z. Wang, Z. Cui, "Low-complexity high-speed decoder design for quasi-cyclic LDPC codes," IEEE Trans. Very Large Scale Integr. Syst. 15, 104-114 (2007).

2006 (1)

Z.-W. Li, L. Chen, L.-Q. Zeng, S. Lin, W. Fong, "Efficient encoding of quasi-cyclic low-density parity-check codes," IEEE Trans. Commun. 54, 71-81 (2006).

2004 (3)

Y. Chen, K. K. Parhi, "Overlapped message passing for quasi-cyclic low-density parity check codes," IEEE Trans. Circuits Syst. I, Reg. Papers 51, 1106-1113 (2004) .

A. Dholakia, E. Eleftheriou, T. Mittelholzer, M. P. C. Fossorier, "Capacity-approaching codes: Can they be applied to the magnetic recording channel?," IEEE Commun. Mag. 42, 122-130 (2004).

R. Tanner, D. Sridhara, A. Sridharan, T. Fuja, D. Costello, Jr., "LDPC block and convolutional codes based on circulant matricies," IEEE Trans. Inf. Theory 50, 2966-2984 (2004).

1999 (1)

D. J. C. MacKay, "Good error-correcting codes based on very sparse matrices," IEEE Trans. Inf. Theory 45, 399-431 (1999).

1997 (1)

D. J. C. MacKay, R. M. Neal, "Near Shannon limit performance of low density parity check codes," Electron. Lett. 33, 457-458 (1997).

1962 (1)

R. G. Gallager, "Low density parity check codes," IRE Trans. Inf. Theory IT-8, 21-28 (1962).

Electron. Lett. (1)

D. J. C. MacKay, R. M. Neal, "Near Shannon limit performance of low density parity check codes," Electron. Lett. 33, 457-458 (1997).

IEEE Commun. Mag. (1)

A. Dholakia, E. Eleftheriou, T. Mittelholzer, M. P. C. Fossorier, "Capacity-approaching codes: Can they be applied to the magnetic recording channel?," IEEE Commun. Mag. 42, 122-130 (2004).

IEEE Trans. Broadcast. (1)

B. Rong, T. Jiang, X. Li, M. R. Soleymani, "Combine LDPC codes over ${\rm GF}(q)$ with $q$-ary modulations for bandwidth-efficient transmission," IEEE Trans. Broadcast. 54, 78-84 (2008).

IEEE Trans. Circuits Syst. I, Reg. Papers (1)

Y. Chen, K. K. Parhi, "Overlapped message passing for quasi-cyclic low-density parity check codes," IEEE Trans. Circuits Syst. I, Reg. Papers 51, 1106-1113 (2004) .

IEEE Trans. Commun. (2)

D. Declercq, M. Fossorier, "Decoding algorithms for nonbinary LDPC codes over ${\rm GF}(q)$," IEEE Trans. Commun. 55, 633-643 (2007).

Z.-W. Li, L. Chen, L.-Q. Zeng, S. Lin, W. Fong, "Efficient encoding of quasi-cyclic low-density parity-check codes," IEEE Trans. Commun. 54, 71-81 (2006).

IEEE Trans. Inf. Theory (3)

D. J. C. MacKay, "Good error-correcting codes based on very sparse matrices," IEEE Trans. Inf. Theory 45, 399-431 (1999).

L. Lan, L. Zeng, Y. Y. Tai, L. Chen, S. Lin, K. Abdel-Ghaffar, "Construction of quasi-cyclic LDPC codes for AWGN and binary erasure channels: A finite field approach," IEEE Trans. Inf. Theory 53, 2429-2458 (2007).

R. Tanner, D. Sridhara, A. Sridharan, T. Fuja, D. Costello, Jr., "LDPC block and convolutional codes based on circulant matricies," IEEE Trans. Inf. Theory 50, 2966-2984 (2004).

IEEE Trans. Very Large Scale Integr. Syst. (1)

Z. Wang, Z. Cui, "Low-complexity high-speed decoder design for quasi-cyclic LDPC codes," IEEE Trans. Very Large Scale Integr. Syst. 15, 104-114 (2007).

IEICE Trans. Commun. (1)

T. Ohtsuki, "LDPC codes in communications and broadcasting," IEICE Trans. Commun. E90-B, 440-453 (2007).

IRE Trans. Inf. Theory (1)

R. G. Gallager, "Low density parity check codes," IRE Trans. Inf. Theory IT-8, 21-28 (1962).

Other (4)

M. Arabaci, I. B. Djordjevic, R. Saunders, R. Marcoccia, "A class of non-binary regular girth-8 LDPC codes for optical communication channels," Proc. OFC/NFOEC 2009 ().

M. C. Davey, Error-correction using low-density parity-check codes Ph.D. dissertation Univ. CambridgeCambridgeU.K. (1999).

S. Lin, D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications (Prentice-Hall, 2004).

C. Spagnol, W. Marnane, E. Popovici, "FPGA implementations of LDPC over ${\rm GF}(2^{m})$ decoders," Proc. IEEE Workshop Signal Process. Syst. (2007) pp. 273-278.

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