Abstract
In this work private information retrieval (PIR) codes are studied. In a k-PIR code, s information bits are encoded in such a way that every information bit has k mutually disjoint recovery sets. The main problem under this paradigm is to minimize the number of encoded bits given the values of s and k, where this value is denoted by P(s, k). The main focus of this work is to analyze P(s, k) for a large range of parameters of s and k. In particular, we improve upon several of the existing results on this value.
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Notes
Note that the definition of a k-PIR code actually depends on the representing generator matrix and not the code itself.
We remark that with respect to lower bounds on N(s, k) it makes also sense to check the entries at http://mint.sbg.ac.at which sometimes contain improvements.
More precisely, it is a strict improvement for \(s=4\) and all \(s\geqslant 7\). An exact formula for N(s, 3), and so also for N(s, 4), exists. It is attained by the Hamming codes and puncturings thereof. The lower bound follows from the Hamming or sphere packing bound.
As an abbreviation set \(k=2^{s-1}\) and number the \(2^{s-1}\) vectors of \(\mathbb {F}_2^s\) with ith component equal to zero by \(x^{1,i},\dots , x^{k,i}\), where we assume that \(x^{1,i}\) is the zero vector. For each \(1\leqslant i\leqslant s\) we can take the recovery sets \(R_1^i=\left\{ x^{1,i}+e_i\right\} \) and \(R_j^i=\left\{ x^{j,i},x^{j,i}+e_i\right\} \) for \(2\leqslant j\leqslant k\).
If the recovery sets have cardinalities in \(\left\{ c_1,\dots ,c_l\right\} \) and there are \(m_i\) recovery sets of cardinality \(c_i\) for \(1\leqslant i\leqslant l\), then we say that the recovery sets have cardinality distribution \(c_1^{m_1} \dots c_l^{m_l}\). The recovery sets for \(e_1,\dots ,e_9\) are given as follows:
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\(e_1\): \(\{0\}\), \(\{1,4,9\}\), \(\{2,16,17\}\), \(\{3,8,19\}\), \(\{5,6,13\}\), \(\{7,21,25\}\), \(\{10,26,27\}\), \(\{11,12,14\}\), \(\{15,20,23\}\), \(\{18,22,24\}\).
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\(e_2\): \(\{0,4,9\}\), \(\{1\}\), \(\{2,5,10\}\), \(\{3,23,24\}\), \(\{6,7,16\}\), \(\{8,13,22\}\), \(\{11,19,21\}\), \(\{12,25,27\}\), \(\{14,15,18\}\), \(\{17,20,26\}\).
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\(e_3\): \(\{0,16,17\}\), \(\{1,5,10\}\), \(\{2\}\), \(\{3,6,12\}\), \(\{4,19,20\}\), \(\{7,8,23\}\), \(\{9,13,14\}\), \(\{11,24,25\}\), \(\{15,22,27\}\), \(\{18,21,26\}\).
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\(e_4\): \(\{0,8,19\}\), \(\{1,23,24\}\), \(\{2,6,12\}\), \(\{3\}\), \(\{4,7,15\}\), \(\{5,9,11\}\), \(\{10,16,18\}\), \(\{13,25,26\}\), \(\{14,20,22\}\), \(\{17,21,27\}\).
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\(e_5\) \(\{0,1,9\}\), \(\{2,19,20\}\), \(\{3,7,15\}\), \(\{4\}\), \(\{5,8,21\}\), \(\{6,10,14\}\), \(\{11,17,18\}\), \(\{12,23,26\}\), \(\{13,24,27\}\), \(\{16,22,25\}\).
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\(e_6\): \(\{0,6,13\}\), \(\{1,2,10\}\), \(\{3,9,11\}\), \(\{4,8,21\}\), \(\{5\}\), \(\{7,12,18\}\), \(\{14,24,26\}\), \(\{15,19,25\}\), \(\{16,20,27\}\), \(\{17,22,23\}\).
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\(e_7\): \(\{0,5,13\}\), \(\{1,7,16\}\), \(\{2,3,12\}\), \(\{4,10,14\}\), \(\{6\}\), \(\{8,15,26\}\), \(\{9,21,22\}\), \(\{11,23,27\}\), \(\{17,19,24\}\), \(\{18,20,25\}\).
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\(e_8\): \(\{0,21,25\}\), \(\{1,6,16\}\), \(\{2,8,23\}\), \(\{3,4,15\}\), \(\{5,12,18\}\), \(\{7\}\), \(\{9,20,24\}\), \(\{10,13,17\}\), \(\{11,22,26\}\), \(\{14,19,27\}\).
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\(e_9\): \(\{0,3,19\}\), \(\{1,13,22\}\), \(\{2,7,23\}\), \(\{4,5,21\}\), \(\{6,15,26\}\), \(\{8\}\), \(\{9,18,27\}\), \(\{10,11,20\}\), \(\{12,16,24\}\), \(\{14,17,25\}\).
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“It can be easily verified that in general \(N(k,t)\leqslant L_P(k,t)\) for small values of \(t>4\). In fact, we will show in Section V that N(k, t) is a tighter lower bound on \(N_P(k,t)\) than \(L_P(k,t)\) for \(r=6\).”.
The value \(N(92,5)=106\) is taken from http://mint.sbg.ac.at.
Note that for \(s=7\) there are two possible cycle types of the cyclic group \(\mathbb {Z}_3\) in \(S_7\) up to conjugation.
The code is even unique within the class of non-linear codes, see [9].
All exhaustive lists of binary linear codes have been enumerated with the software package LinCode, see [16].
We refer to a doubled Reed-Muller code as the code generate by concatenating a generator of a Reed-Muller code twice with itself.
Note that this can be slightly ambiguous, if the exact value was determined. In [18] the upper bound \(P(5,8)\leqslant 18\) was obtained and we only improved the lower bound.
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Appendix A: Generator Matrices for the Stated Best Known Upper Bounds on P(s, k)
Appendix A: Generator Matrices for the Stated Best Known Upper Bounds on P(s, k)
In this section we state generator matrix for the currently best known codes found by the ILP approach from Sect. 4, where we only consider the cases of dimension \(s\geqslant 2\) and demand \(k\geqslant 4\). Note that all codes are projective and except for the cases of P(6, 8), P(6, 10), P(6, 12), P(7, 12), P(9, 16) all generator matrices are systematic.
\(P(4,4)\leqslant 9\)
\(P(4,6)\leqslant 12\)
\(P(5,4)\leqslant 10\)
\(P(5,6)\leqslant 14\)
\(P(5,8)\leqslant 18\)
\(P(5,10)\leqslant 22\)
\(P(5,12)\leqslant 25\)
\(P(5,14)\leqslant 28\)
\(P(6,4)\leqslant 11\)
\(P(6,6)\leqslant 15\)
\(P(6,8)\leqslant 19\)
\(P(6,10)\leqslant 23\)
\(P(6,12)\leqslant 27\)
\(P(6,14)\leqslant 32\)
\(P(6,16)\leqslant 36\)
\(P(7,6)\leqslant 16\)
\(P(7,8)\leqslant 21\)
\(P(7,10)\leqslant 26\)
\(P(7,12)\leqslant 29\)
\(P(7,14)\leqslant 34\)
\(P(7,16)\leqslant 39\)
\(P(7,32)\leqslant 71\)
\(P(8,6)\leqslant 18\)
\(P(8,8)\leqslant 23\)
\(P(8,10)\leqslant 27\)
\(P(8,12)\leqslant 33\)
\(P(8,14)\leqslant 38\)
\(P(8,16)\leqslant 42\)
\(P(9,10)\leqslant 28\)
\(P(9,12)\leqslant 37\)
\(P(9,14)\leqslant 40\)
\(P(9,16)\leqslant 45\)
\(P(10,10)\leqslant 31\)
\(P(10,12)\leqslant 40\)
\(P(10,14)\leqslant 45\)
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Kurz, S., Yaakobi, E. PIR Codes with Short Block Length. Des. Codes Cryptogr. 89, 559–587 (2021). https://doi.org/10.1007/s10623-020-00828-6
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DOI: https://doi.org/10.1007/s10623-020-00828-6