Cryptography and Communications

, Volume 8, Issue 1, pp 19–32 | Cite as

On encoding symbol degrees of array BP-XOR codes

  • Maura B. Paterson
  • Douglas R. Stinson
  • Yongge Wang


Low density parity check (LDPC) codes, LT codes and digital fountain techniques have received significant attention from both academics and industry in the past few years. By employing the underlying ideas of efficient Belief Propagation (BP) decoding process (also called iterative message passing decoding process) on binary erasure channels (BEC) in LDPC codes, Wang has recently introduced the concept of array BP-XOR codes and showed the necessary and sufficient conditions for MDS [k + 2,k] and [n,2] array BP-XOR codes. In this paper, we analyze the encoding symbol degree requirements for array BP-XOR codes and present new necessary conditions for array BP-XOR codes. These new necessary conditions are used as a guideline for constructing several array BP-XOR codes and for presenting a complete characterization (necessary and sufficient conditions) of degree two array BP-XOR codes and for designing new edge-colored graphs. Meanwhile, these new necessary conditions are used to show that the codes by Feng, Deng, Bao, and Shen in IEEE Transactions on Computers are incorrect.


Array codes Encoding symble degrees MDS array codes Bounds on codes Error corecting codes 

Mathematics Subject Classification (2010)

94B65 94A45 94B05 



The authors would like to thank the anonymous reviewers for detailed comments on improving the presentation of this paper.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Maura B. Paterson
    • 1
  • Douglas R. Stinson
    • 2
  • Yongge Wang
    • 3
  1. 1.Department Economics, Mathematics and StatisticsBirkbeck University of LondonLondonUK
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  3. 3.Department of Software and Information SystemsUNC CharlotteCharlotteUSA

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