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Binary, Shortened Projective Reed Muller Codes for Coded Private Information Retrieval

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 Added by Myna Vajha
 Publication date 2017
and research's language is English




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The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with replicated-server PIR. In the present paper, the construction of an $(n,k)$ $tau$-server binary, linear PIR code having parameters $n = sumlimits_{i = 0}^{ell} {m choose i}$, $k = {m choose ell}$ and $tau = 2^{ell}$ is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary, Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any $tau$ of the form $2^{ell}$, $2^{ell}-1$ and any value of $k$, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with other PIR codes appearing in the literature. For the particular case of $tau=3,4$, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length $n(k, tau)$ of a systematic PIR code. It follows from a result by Vardy and Yaakobi, that these codes also yield optimal, systematic primitive multi-set $(n, k, tau)_B$ batch codes for $tau=3,4$. The PIR code constructions presented here also yield upper bounds on the generalized Hamming weights of binary PRM codes.



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Projective Reed-Muller codes were introduced by Lachaud, in 1988 and their dimension and minimum distance were determined by Serre and S{o}rensen in 1991. In coding theory one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as next-to-minimal weight) is completely determined. In this paper we determine all the values of the next-to-minimal weight for the binary projective Reed-Muller codes, which we show to be equal to the next-to-minimal weight of Reed-Muller codes in most, but not all, cases.
A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.
We consider the problem of Private Information Retrieval with Private Side Information (PIR-PSI), wherein a user wants to retrieve a file from replication based non-colluding databases by using the prior knowledge of a subset of the files stored on the databases. The PIR-PSI framework ensures that the privacy of the demand and the side information are jointly preserved, thereby finding potential applications when multiple files have to be downloaded spread across different time-instants. Although the capacity of the PIR-PSI setting is known, we observe that the underlying capacity achieving code construction uses Maximum Distance Separable (MDS) codes thereby contributing to high computational complexity when retrieving the demand. Pointing at this drawback of MDS-based PIR-PSI codes, we propose XOR-based PIR-PSI codes for a simple yet non-trivial setting of two non-colluding databases and two side information files at the user. While our codes offer substantial reduction in complexity when compared to MDS based codes, the code-rate marginally falls short of the capacity of the PIR-PSI setting. Nevertheless, we show that our code-rate is strictly higher than that of XOR-based codes for PIR with no side information, thereby implying that our codes can be useful when downloading multiple files in a sequential manner, instead of applying XOR-based PIR codes on each file.
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The well known Plotkin construction is, in the current paper, generalized and used to yield new families of Z2Z4-additive codes, whose length, dimension as well as minimum distance are studied. These new constructions enable us to obtain families of Z2Z4-additive codes such that, under the Gray map, the corresponding binary codes have the same parameters and properties as the usual binary linear Reed-Muller codes. Moreover, the first family is the usual binary linear Reed-Muller family.
The famous Barnes-Wall lattices can be obtained by applying Construction D to a chain of Reed-Muller codes. By applying Construction ${{D}}^{{(cyc)}}$ to a chain of extended cyclic codes sandwiched between Reed-Muller codes, Hu and Nebe (J. London Math. Soc. (2) 101 (2020) 1068-1089) constructed new series of universally strongly perfect lattices sandwiched between Barnes-Wall lattices. In this paper, we explicitly determine the minimum weight codewords of those codes for some special cases.
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