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Message encoding and retrieval for spread and cyclic orbit codes

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Abstract

Spread codes and cyclic orbit codes are special families of constant dimension subspace codes. These codes have been well-studied for their error correction capability, transmission rate and decoding methods, but the question of how to encode and retrieve messages has not been investigated. In this work we show how a message set of consecutive integers can be encoded and retrieved for these two code families.

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Notes

  1. As shown in the proof of Lemma 17, for \(j=2,\dots ,k\) the j-th row of \(\psi _k(P^\ell )\) is the \(\alpha ^{j-1}\)-multiple of the first row. Analogously, the same statement holds for any element of \(\mathbb {F}_q[P]\). It follows that all non-zero elements of the codeword \({\mathcal {U}}\), when represented in \(\mathbb {F}_{q^k}^m\), are \(\mathbb {F}_{q^k}\)-multiples of each other.

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Acknowledgements

The author would like to thank Yuval Cassuto for his reference to enumerative coding, John Sheekey for his advice on Desarguesian spreads, and Margreta Kuijper for fruitful discussions and comments on this work. The author was partially supported by Swiss National Science Foundation Fellowship No. 147304.

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  1. Faculty of Mathematics and Statistics, University of St. Gallen, St. Gallen, Switzerland

    Anna-Lena Horlemann-Trautmann

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  1. Anna-Lena Horlemann-Trautmann
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Correspondence to Anna-Lena Horlemann-Trautmann.

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This is one of several papers published in Designs, Codes and Cryptography comprising the Special Issue on Network Coding and Designs.

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Horlemann-Trautmann, AL. Message encoding and retrieval for spread and cyclic orbit codes. Des. Codes Cryptogr. 86, 365–386 (2018). https://doi.org/10.1007/s10623-017-0377-x

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