| Paper ID | D7-S3-T4.2 |
| Paper Title |
Almost Optimal Construction of Functional Batch Codes Using Hadamard Codes |
| Authors |
Lev Yohananov, Eitan Yaakobi, Technion, Israel |
| Session |
D7-S3-T4: Batch Codes |
| Chaired Session: |
Tuesday, 20 July, 22:40 - 23:00 |
| Engagement Session: |
Tuesday, 20 July, 23:00 - 23:20 |
| Abstract |
A \textit{functional $k$-batch} code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^s-1$ servers. This conjecture is verified for $s\leq 5$ but previous work could only show that codes with $n=2^s-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left\lfloor \frac{ 2^{s/2}}{\sqrt{24}} \right\rfloor$ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac{2}{3}2^{s-1} \rfloor$ requests with the same number of servers. %$n=2^s-1$ servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result. %We provide some bounds on the minimum number of servers for functional $k$-batch codes. These constructions are mainly based on Hadamard codes and equivalently provide constructions for \textit{parallel Random I/O (RIO)} codes.
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