Query-Efficient Locally Decodable Codes of Subexponential Length

We develop the algebraic theory behind the constructions of Yekhanin (2008) and Efremenko (2009), in an attempt to understand the ``algebraic niceness’’ phenomenon in $\mathbb{Z}m$. We show that every integer $m = pq = 2^t -1$, where $p$, $q$ and $t$ are prime, possesses the same good algebraic property as $m=511$ that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki’s composition method. More precisely, we construct a $3^{\lceil r/2\rceil}$-query LDC for every positive integer $r<104$ and a $\left\lfloor (3/4)^{51}\cdot 2^{r}\right\rfloor$-query LDC for every integer $r\geq 104$, both of length $N{r}$, improving the $2^r$ queries used by Efremenko (2009) and $3\cdot 2^{r-2}$ queries used by Itoh and Suzuki (2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.

💡 Research Summary

The paper addresses a central problem in the theory of locally decodable codes (LDCs): how to reduce the number of queries required to recover any single message symbol while keeping the code length subexponential in the message length. The authors build on the breakthrough constructions of Yekhanin (2008) and Efremenko (2009), which introduced the notion of “algebraic niceness” of certain moduli $m$ in the ring $\mathbb{Z}_m$. In those works a special modulus $m=511$ (which equals $7\cdot73=2^9-1$) was shown to admit a very efficient three‑query decoding step that can be recursively composed, yielding LDCs with query complexity $3^{\lceil r/2\rceil}$ for a parameter $r$ that controls the code length $N_r$.

The present work first generalizes this phenomenon. The authors prove that any integer of the form
\