Low-Density Permutation Codes for Digital Holographic Data Storage

1. Low-Density Permutation Codes for Digital Holographic Data Storage Sergei S. Orlov Stanford University Kirill V. Shcheglov NASA Jet Propulsion Laboratory Hongtao Liu, Snezhana I. Abarzhi Illinois Institute of Technology

2. Motivation • High total code rate (channel and ECC combined) • Error correction through sparse encoding • Most efficient, Maximum Likelihood detection

3. Outline • Channel coding in holographic data storage • Channel constraint coding with strong error correction • Maximum likelihood detection algorithm, NP complete to solve, exactly evaluated at nlog(n) • Strict Hamming distance codes versus pseudo-random encoding schemes: low SNR limit • Performance of maximum likelihood permutation codes versus Shannon limit

4. Basic holographic data storage system and its components

5. Information coding in holographic storage: Hamming constraint block coding (e.g., 6 to 8 HDSS demo) Block coding : each 2 x 4 = 8 pixels encodes 6 bits of ECC+User information 1024 × 1024 pixels image

6. Example: 1 Mpixel holographic data page • Digitized high-resolution (1024x1024 pixels; 13.1 x 13.1 mm) holographic image captured at 1,000 frames-per-second • Each pixel in this image is a unique data channel

7. Digital holographic storage system, Stanford’2000 Pulsed Nd:YAG laser (not shown) Double Fourier transform lens Kodak C7 1000 fps CCD IBM FLC 640 fps SLM Optical shaft encoder Air-bearing spindle Polaroid / Aprilis photopolymer disk

8. Digital signal processing schematic global threshold vs. constraint block coding Signal-to-Noise Ratio map Performance critically depends on the global threshold value

9. Approach: Sparse source coding with strict maximum likelihood detection

10. Encoding scheme: enumeration algorithm by Cover

11. 1024 512 256 N = 32 64 128 Coding rate versus block size and sparseness • High code rate and high error redundancy (large M) can be realized with sufficiently large block sizes

12. Channel constraint codingwith strong error correction (BER < 10-10 or less)

13. Generic properties of sparse permutation codes • Asymptotic block error rate: • For AWGN channel: • Number of iterations:

14. Error Statistics by Hamming distance At low SNR most errors occur into the codewords of large Hamming distance. Regularized fixed Hamming distance coding can not guarantee “zero” error probability at reasonable code rates. Pseudo-random encoding with the same effective sparseness is as efficient as the strict Hamming distance coding

15. Performance of permutation channel codes in the low SNR limit • BER  510-3 at SNR ~2.3 (uncoded bit-error-rate of ~5%) • Combined with high ~0.9 code rate TPC or LDPC • BER < 10-12 at SNR ~2.4 and Code rate > 0.55

16. Channel capacity and performance of permutation codes with different block sizes Shannon limit for binary signaling: N→ ∞, BER→ 0 • Codes with sufficiently large block sizes essentially approach the theoretical Shannon capacity • Small block size codes used previously are inefficient. Minimum block size to realize the capacity limit: N ~ >50 (depends on the target BER)

17. Summary and Conclusions • Channel constraint coding with simultaneous strong error correction demonstrated • Exact Maximum Likelihood detection realized with large block sizes using a new iterative decoding algorithm • Code rate performance of sufficiently large block size codes closely approach the fundamental channel capacity limits • Experimental verification and demonstration of the codes performance planned with Aprilis, Inc. holographic platform