Improved power decoding of interleaved one-point Hermitian codes

Abstract

An \(h\)-interleaved one-point Hermitian code is a direct sum of \(h\) many one-point Hermitian codes, where errors are assumed to occur at the same positions in the constituent codewords. We propose a new partial decoding algorithm for these codes that can decode—under certain assumptions—an error of relative weight up to \(1-\big (\tfrac{k+g}{n}\big )^{\frac{h}{h+1}}\), where k is the dimension, n the length, and g the genus of the code. Simulation results for various parameters indicate that the new decoder achieves this maximal decoding radius with high probability. The algorithm is based on a recent generalization of improved power decoding to interleaved Reed–Solomon codes, does not require an expensive root-finding step, and improves upon the previous best decoding radius at all rates. In the special case \(h=1\), we obtain an adaption of the improved power decoding algorithm to one-point Hermitian codes, which for all simulated parameters achieves a similar observed failure probability as the Guruswami–Sudan decoder above the latter’s guaranteed decoding radius.

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Fig. 1

Notes

  1. 1.

    Over \({\mathbb {F}}_{q^2}[X]\) we would have \(\varLambda _s=\varLambda ^s\). Over \({\mathcal {R}}\), however, \(\deg _{{\mathcal {H}}}\varLambda _s \le s|{\mathcal {E}}| + g\) while we would often have \(\deg _{{\mathcal {H}}}\varLambda ^s = s(|{\mathcal {E}}|+g)\).

  2. 2.

    For \(s=1\), the fraction of error vectors of weight \(|{\mathcal {E}}|\) with \(\deg _{{\mathcal {H}}}\varLambda _1 < |{\mathcal {E}}| + g\) is approximately \(1-\tfrac{1}{q^2}\), cf. [8, 9]. Our numerical results indicate that a similar statement holds for \(s>1\).

  3. 3.

    There are rare cases in which \(\deg _{{\mathcal {H}}}(\sum _{{{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}}\in {\mathcal {I}}} \lambda _{{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}}A_{{{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}},{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}}}) < \tau + {|{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}}|} (n+2g-1)\) (cf. Remark 2) for which the number of equations is smaller.

  4. 4.

    This linear-algebraic condition resembles, but seems weaker than, the “(non-linear) algebraic independence assumption” in [5] for decoding interleaved RS codes.

  5. 5.

    As stated, the bound holds for arbitrary \(\ell \). However, it results in values greater whenever the number of errors s larger than the decoding radius of having parameter \(\ell = 2\).

  6. 6.

    The problem discussed in [16, Sect. V.B] consists of a system of congruences with degree constraints. Since Problem 2 involves both equations (\(1 \le {|{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}}|} < s\)) and congruences (\(s \le {|{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}}|} \le \ell \)), w.l.o.g., we first need to rewrite the equations into congruences modulo \(x^\xi \), where \(\xi \) is greater than the largest possible degree of the left- and right-hand side of the equations (i.e., \(\psi _{{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}},\kappa }\) and \(\sum _{{{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}}\in {\mathcal {I}}} \sum _{\iota =0}^{q-1} \lambda _{{{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}},\iota } A^{({{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}},{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}})}_{\iota ,\kappa }\)) for a solution \(\lambda _{{{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}},\iota }\) and \(\psi _{{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}},\kappa }\) of minimal degree. Upper bounds on these degrees can be obtained using the solutions \({{\mathchoice{{\varvec{\displaystyle \nu }}}{{\varvec{\textstyle \nu }}}{{\varvec{\scriptstyle \nu }}}{{\varvec{\scriptscriptstyle \nu }}}}}(\varLambda _{{\mathchoice{{\varvec{\displaystyle i}}}{{\varvec{\textstyle i}}}{{\varvec{\scriptstyle i}}}{{\varvec{\scriptscriptstyle i}}}}})\) and \({{\mathchoice{{\varvec{\displaystyle \nu }}}{{\varvec{\textstyle \nu }}}{{\varvec{\scriptstyle \nu }}}{{\varvec{\scriptscriptstyle \nu }}}}}(\varPsi _{{\mathchoice{{\varvec{\displaystyle j}}}{{\varvec{\textstyle j}}}{{\varvec{\scriptstyle j}}}{{\varvec{\scriptscriptstyle j}}}}})\) (cf. Theorem 3) corresponding to the actual error locator of multiplicity s and the maximal number of errors that we can correct (cf. Sect. 5).

  7. 7.

    As pointed out in [24], an RS code over \({\mathbb {F}}_{q^{6h}}\) whose evaluation points all lie in \({\mathbb {F}}_{q^3}\) are equivalent to a 2h-interleaved RS codes over \({\mathbb {F}}_{q^3}\), i.e. can be decoded up to \(t_{\mathrm {IRS}}\). In our comparison here we therefore consider RS codes with arbitrary evaluation points for which this equivalence doesn’t hold.

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Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments, which improved the readability of the paper.

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Puchinger, S., Rosenkilde, J. & Bouw, I. Improved power decoding of interleaved one-point Hermitian codes. Des. Codes Cryptogr. 87, 589–607 (2019). https://doi.org/10.1007/s10623-018-0577-z

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Keywords

  • Interleaved one-point Hermitian codes
  • Power decoding
  • Collaborative decoding

Mathematics Subject Classification

  • 94B35
  • 14G50