On modules with cyclic socle

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.

Limitations of single coset states and quantum algorithms for code equivalence

Quantum computers can break the RSA, El Gamal, and elliptic curve public-key cryptosystems, as they can efficiently factor integers and extract discrete logarithms. The power of such quantum attacks lies in \emph{quantum Fourier sampling}, an algorithmic paradigm based on generating and measuring coset states. %This motivates the investigation of the power or limitations of quantum Fourier sampling, especially in attacking candidates for ``post-quantum'' cryptosystems -- classical cryptosystems that can be implemented with today's computers but will remain secure even in the presence of quantum attacks. In this article we extend previous negative results of quantum Fourier sampling for Graph Isomorphism, which corresponds to hidden subgroups of order two (over S_n, to several cases corresponding to larger hidden subgroups. For one case, we strengthen some results of Kempe, Pyber, and Shalev on the Hidden Subgroup Problem over the symmetric group. In another case, we show the failure of quantum Fourier sampling on the Hidden Subgroup Problem over the general linear group GL_2(\FF_q). The most important case corresponds to Code Equivalence, the problem of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. Our results suggest that for many codes of interest---including generalized Reed Solomon codes, alternant codes, and Reed-Muller codes---solving these instances of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms.