SPC product codes, graphs with cycles and Kostka numbers

  • Sara D. CardellEmail author
  • Joan-Josep Climent
  • Alberto López Martín
Original Paper


The SPC product code is a very popular error correction code with four as its minimum distance. Over the erasure channel, it is supposed to correct up to three erasures. However, this code can correct a higher number of erasures under certain conditions. A codeword of the SPC product code can be represented either by an erasure pattern or by a bipartite graph, where the erasures are represented by an edge. When the erasure contains erasures that cannot be corrected, the corresponding graph contains cycles. In this work we determine the number of strict uncorrectable erasure patterns (bipartite graphs with cycles) for a given size with a fixed number of erasures (edges). Since a bipartite graph can be unequivocally represented by its biadjacency matrix, it is enough to determine the number of non-zero binary matrices whose row and column sum vectors are different from one. At the same time, the number of matrices with prescribed row and column sum vectors can be evaluated in terms of the Kostka numbers associated with Young tableaux.


SPC code Erasure channel Erasure pattern Bipartite graph Kostka number 

Mathematics Subject Classification

94B05 05C30 



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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Instituto de Matemática, Estatística e Computação CientíficaUniversity of CampinasCampinasBrazil
  2. 2.Departament de MatemàtiquesUniversitat d’AlacantAlacantSpain
  3. 3.Instituto Nacional de Matemática Pura e AplicadaRio de JaneiroBrazil

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