Abstract

We investigate the application of irregular repeat-accumulate (IRA) codes in volume holographic memory (VHM) systems. We introduce methodologies to design efficient IRA codes. We show that a judiciously designed IRA code for a typical VHM can be as good as the optimized irregular low-density-parity-check codes while having the additional advantage of lower encoding complexity. Moreover, we present a method to reduce the error-floor effect of the IRA codes in the VHM systems. This method explores the structure of the noise pattern in holographic memories. Finally, we explain why IRA codes are good candidates for the VHM systems.

© 2004 Optical Society of America

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References

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  1. W. Chou, M. A. Neifeld, “Interleaving and error correction in volume holographic memory systems,” Appl. Opt. 37, 6951–6968 (1998).
    [CrossRef]
  2. M. A. Neifeld, M. McDonald, “Error correction for increasing the usable capacity photorefractive memories,” Opt. Lett. 19, 1483–1485 (1994).
    [CrossRef] [PubMed]
  3. B. J. Geortzen, P. A. Mitkas, “Error-correcting code for volume holographic storage of a relational database,” Opt. Lett. 20, 1655–1657 (1995).
    [CrossRef]
  4. G. W. Burr, J. Ashley, H. Coufal, R. K. Grygier, J. A. Hoffnagle, C. M. Jefferson, B. Marcus, “Modulation coding for pixel-matched holographic data storage,” Opt. Lett. 22, 639–641 (1997).
    [CrossRef] [PubMed]
  5. M. A. Neifeld, W. Chou, “Information theoretic limits to the capacity of volume holographic optical memory,” Appl. Opt. 36, 514–517 (1997).
    [CrossRef] [PubMed]
  6. W. Chou, M. A. Neifeld, “Soft-decision array decoding for volume holographic memory systems,” J. Opt. Soc. Am. A 18, 185–194 (2001).
    [CrossRef]
  7. H. Pishro-Nik, N. Rahnavard, J. Ha, F. Fekri, A. Adibi, “Low-density parity-check codes for volume holographic memory systems,” Appl. Opt. 42, 861–870 (2003).
    [CrossRef] [PubMed]
  8. T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 638–656 (2001).
    [CrossRef]
  9. A. Kavčić, X. Ma, M. Mitzenmacher, “Binary inersymbol interference channels: Gallager codes, density evolution and code performance bounds,” IEEE Trans. Inf. Theory 49, 1636–1652 (2003).
    [CrossRef]
  10. H. Jin, A. Khandekar, R. McEliece, “Irregular repat-accumulate codes,” presented at the Second International Symposium on Turbo Codes and Related Topics, Brest, France, 4–7 September 2000.
  11. R. G. Galleger, Low-density Parity-Check Codes (MIT Press, Cambridge, Mass., 1963).
  12. D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45, 399–431 (1999).
    [CrossRef]
  13. M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory 47, 585–598 (2001).
    [CrossRef]
  14. T. J. Richardson, R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
    [CrossRef]
  15. T. J. Richardson, M. A. Shokrollahi, R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 619–637 (2001).
    [CrossRef]
  16. H. Pishro-Nik, N. Rahnavard, F. Fekri, “Nonuniform error correction using low-density parity check codes,” in Proceedings of Fortieth Annual Allerton Conference (University of Illinois at Urbana-Champaign, Champaign, Ill., 2002), available on CD-ROM.
  17. H. Pishro-Nik, N. Rahnavard, F. Fekri, “Results on non-uniform error correction using low-density parity-check codes,” in Global Telecommunications Conference (Institute of Electrical and Electronics Engineers, New York, 2003), available on CD-ROM.
  18. M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory 47, 569–584 (2001).
    [CrossRef]
  19. C. Di, T. Richardson, R. Urbanke, “Weight distributions: How deviant can you be?,” in 2001 IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, New York, 2001), p. 50.
  20. A. Orlitskey, K. Viswanathan, J. Zhang, “Stopping set distribution of ldpc code ensembles,” IEEE Trans. Inf. Theory (to be published).
  21. D. J. Brady, D. Psaltis, “Control of volume holograms,” J. Opt. Soc. Am. A 9, 1167–1182 (1992).
    [CrossRef]
  22. J. P. Drolet, E. Chuang, G. Barbastathis, D. Psaltis, “Compact, integrated dynamic holographic memory with refreshed holograms,” Opt. Lett. 22, 552–554 (1997).
    [CrossRef] [PubMed]
  23. Y. Owechko, “Cascaded-grating holography for artificial neural networks,” Appl. Opt. 32, 1380–1398 (1993).
    [CrossRef] [PubMed]

2003

H. Pishro-Nik, N. Rahnavard, J. Ha, F. Fekri, A. Adibi, “Low-density parity-check codes for volume holographic memory systems,” Appl. Opt. 42, 861–870 (2003).
[CrossRef] [PubMed]

A. Kavčić, X. Ma, M. Mitzenmacher, “Binary inersymbol interference channels: Gallager codes, density evolution and code performance bounds,” IEEE Trans. Inf. Theory 49, 1636–1652 (2003).
[CrossRef]

2001

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory 47, 585–598 (2001).
[CrossRef]

T. J. Richardson, R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[CrossRef]

T. J. Richardson, M. A. Shokrollahi, R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 619–637 (2001).
[CrossRef]

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory 47, 569–584 (2001).
[CrossRef]

T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 638–656 (2001).
[CrossRef]

W. Chou, M. A. Neifeld, “Soft-decision array decoding for volume holographic memory systems,” J. Opt. Soc. Am. A 18, 185–194 (2001).
[CrossRef]

1999

D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45, 399–431 (1999).
[CrossRef]

1998

1997

1995

1994

1993

1992

Adibi, A.

Ashley, J.

Barbastathis, G.

Brady, D. J.

Burr, G. W.

Chou, W.

Chuang, E.

Coufal, H.

Di, C.

C. Di, T. Richardson, R. Urbanke, “Weight distributions: How deviant can you be?,” in 2001 IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, New York, 2001), p. 50.

Drolet, J. P.

Fekri, F.

H. Pishro-Nik, N. Rahnavard, J. Ha, F. Fekri, A. Adibi, “Low-density parity-check codes for volume holographic memory systems,” Appl. Opt. 42, 861–870 (2003).
[CrossRef] [PubMed]

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Nonuniform error correction using low-density parity check codes,” in Proceedings of Fortieth Annual Allerton Conference (University of Illinois at Urbana-Champaign, Champaign, Ill., 2002), available on CD-ROM.

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Results on non-uniform error correction using low-density parity-check codes,” in Global Telecommunications Conference (Institute of Electrical and Electronics Engineers, New York, 2003), available on CD-ROM.

Galleger, R. G.

R. G. Galleger, Low-density Parity-Check Codes (MIT Press, Cambridge, Mass., 1963).

Geortzen, B. J.

Grygier, R. K.

Ha, J.

Hoffnagle, J. A.

Jefferson, C. M.

Jin, H.

H. Jin, A. Khandekar, R. McEliece, “Irregular repat-accumulate codes,” presented at the Second International Symposium on Turbo Codes and Related Topics, Brest, France, 4–7 September 2000.

Kavcic, A.

A. Kavčić, X. Ma, M. Mitzenmacher, “Binary inersymbol interference channels: Gallager codes, density evolution and code performance bounds,” IEEE Trans. Inf. Theory 49, 1636–1652 (2003).
[CrossRef]

Khandekar, A.

H. Jin, A. Khandekar, R. McEliece, “Irregular repat-accumulate codes,” presented at the Second International Symposium on Turbo Codes and Related Topics, Brest, France, 4–7 September 2000.

Luby, M.

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory 47, 585–598 (2001).
[CrossRef]

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory 47, 569–584 (2001).
[CrossRef]

Ma, X.

A. Kavčić, X. Ma, M. Mitzenmacher, “Binary inersymbol interference channels: Gallager codes, density evolution and code performance bounds,” IEEE Trans. Inf. Theory 49, 1636–1652 (2003).
[CrossRef]

MacKay, D. J. C.

D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45, 399–431 (1999).
[CrossRef]

Marcus, B.

McDonald, M.

McEliece, R.

H. Jin, A. Khandekar, R. McEliece, “Irregular repat-accumulate codes,” presented at the Second International Symposium on Turbo Codes and Related Topics, Brest, France, 4–7 September 2000.

Mitkas, P. A.

Mitzenmacher, M.

A. Kavčić, X. Ma, M. Mitzenmacher, “Binary inersymbol interference channels: Gallager codes, density evolution and code performance bounds,” IEEE Trans. Inf. Theory 49, 1636–1652 (2003).
[CrossRef]

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory 47, 585–598 (2001).
[CrossRef]

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory 47, 569–584 (2001).
[CrossRef]

Neifeld, M. A.

Orlitskey, A.

A. Orlitskey, K. Viswanathan, J. Zhang, “Stopping set distribution of ldpc code ensembles,” IEEE Trans. Inf. Theory (to be published).

Owechko, Y.

Pishro-Nik, H.

H. Pishro-Nik, N. Rahnavard, J. Ha, F. Fekri, A. Adibi, “Low-density parity-check codes for volume holographic memory systems,” Appl. Opt. 42, 861–870 (2003).
[CrossRef] [PubMed]

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Results on non-uniform error correction using low-density parity-check codes,” in Global Telecommunications Conference (Institute of Electrical and Electronics Engineers, New York, 2003), available on CD-ROM.

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Nonuniform error correction using low-density parity check codes,” in Proceedings of Fortieth Annual Allerton Conference (University of Illinois at Urbana-Champaign, Champaign, Ill., 2002), available on CD-ROM.

Psaltis, D.

Rahnavard, N.

H. Pishro-Nik, N. Rahnavard, J. Ha, F. Fekri, A. Adibi, “Low-density parity-check codes for volume holographic memory systems,” Appl. Opt. 42, 861–870 (2003).
[CrossRef] [PubMed]

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Nonuniform error correction using low-density parity check codes,” in Proceedings of Fortieth Annual Allerton Conference (University of Illinois at Urbana-Champaign, Champaign, Ill., 2002), available on CD-ROM.

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Results on non-uniform error correction using low-density parity-check codes,” in Global Telecommunications Conference (Institute of Electrical and Electronics Engineers, New York, 2003), available on CD-ROM.

Richardson, T.

C. Di, T. Richardson, R. Urbanke, “Weight distributions: How deviant can you be?,” in 2001 IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, New York, 2001), p. 50.

Richardson, T. J.

T. J. Richardson, R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[CrossRef]

T. J. Richardson, M. A. Shokrollahi, R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 619–637 (2001).
[CrossRef]

T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 638–656 (2001).
[CrossRef]

Shokrollahi, M.

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory 47, 585–598 (2001).
[CrossRef]

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory 47, 569–584 (2001).
[CrossRef]

Shokrollahi, M. A.

T. J. Richardson, M. A. Shokrollahi, R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 619–637 (2001).
[CrossRef]

Spielman, D.

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory 47, 585–598 (2001).
[CrossRef]

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory 47, 569–584 (2001).
[CrossRef]

Urbanke, R.

C. Di, T. Richardson, R. Urbanke, “Weight distributions: How deviant can you be?,” in 2001 IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, New York, 2001), p. 50.

Urbanke, R. L.

T. J. Richardson, M. A. Shokrollahi, R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 619–637 (2001).
[CrossRef]

T. J. Richardson, R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[CrossRef]

T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 638–656 (2001).
[CrossRef]

Viswanathan, K.

A. Orlitskey, K. Viswanathan, J. Zhang, “Stopping set distribution of ldpc code ensembles,” IEEE Trans. Inf. Theory (to be published).

Zhang, J.

A. Orlitskey, K. Viswanathan, J. Zhang, “Stopping set distribution of ldpc code ensembles,” IEEE Trans. Inf. Theory (to be published).

Appl. Opt.

IEEE Trans. Inf. Theory

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory 47, 569–584 (2001).
[CrossRef]

D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory 45, 399–431 (1999).
[CrossRef]

M. Luby, M. Mitzenmacher, M. Shokrollahi, D. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory 47, 585–598 (2001).
[CrossRef]

T. J. Richardson, R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[CrossRef]

T. J. Richardson, M. A. Shokrollahi, R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 619–637 (2001).
[CrossRef]

T. J. Richardson, R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” IEEE Trans. Inf. Theory 47, 638–656 (2001).
[CrossRef]

A. Kavčić, X. Ma, M. Mitzenmacher, “Binary inersymbol interference channels: Gallager codes, density evolution and code performance bounds,” IEEE Trans. Inf. Theory 49, 1636–1652 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

H. Jin, A. Khandekar, R. McEliece, “Irregular repat-accumulate codes,” presented at the Second International Symposium on Turbo Codes and Related Topics, Brest, France, 4–7 September 2000.

R. G. Galleger, Low-density Parity-Check Codes (MIT Press, Cambridge, Mass., 1963).

C. Di, T. Richardson, R. Urbanke, “Weight distributions: How deviant can you be?,” in 2001 IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, New York, 2001), p. 50.

A. Orlitskey, K. Viswanathan, J. Zhang, “Stopping set distribution of ldpc code ensembles,” IEEE Trans. Inf. Theory (to be published).

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Nonuniform error correction using low-density parity check codes,” in Proceedings of Fortieth Annual Allerton Conference (University of Illinois at Urbana-Champaign, Champaign, Ill., 2002), available on CD-ROM.

H. Pishro-Nik, N. Rahnavard, F. Fekri, “Results on non-uniform error correction using low-density parity-check codes,” in Global Telecommunications Conference (Institute of Electrical and Electronics Engineers, New York, 2003), available on CD-ROM.

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Figures (4)

Fig. 1
Fig. 1

Tanner graph representation of the codes from the ensemble defined by (f 1, f 2, …, f J ; a).

Fig. 2
Fig. 2

Reducing the error-floor effect of IRA codes in VHM systems.

Fig. 3
Fig. 3

Comparison of different coding schemes for a VHM system.

Fig. 4
Fig. 4

Performance of an IRA code of length 104 and rate 0.9.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

xj=xj-1+i=1a vj-1a+i,
xj+xj-1+i=1a vj-1a+i=0.
λjx= λijxi-1,
λij=EijEj.
x0j=j, z0=kr.
xl=j qjxlj,
yl=1-1-zl-1R1-xl-1,
zl=z0yl,
wl=1-1-zl-12ρ1-xl-1,
xlj=jλjwl,
Rx=0x ρtdt01 ρtdt.
H=H1|H2,
H2=100000011000000110000000110000001100000011.
C=- ϕxlog2ϕxdx-12log22πeσ2,
ϕx=18πσ2exp-x+12/2σ2+exp-x-12/2σ2,
SNR2-SNR1=1.61 dB, SNR3-SNR1=2.80 dB, SNR4-SNR1=3.74 dB.

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