Faster construction of asymptotically good unit-cost error correcting codes in the RAM model

TL;DR

This paper presents a novel method for constructing error-correcting codes in the RAM model that can be built faster than linear in word size, enabling constant-time encoding for w-bit numbers.

Contribution

It introduces a construction of error-correcting codes with constant relative distance that can be built in sublinear time in the word size w, improving previous exponential-time methods.

Findings

Construction time is o(w), faster than previous exponential methods.

Encoding of w-bit numbers is achievable in constant time.

Codes have constant relative positive distance and map w-bit numbers to Θ(w)-bit numbers.

Abstract

Assuming we are in a Word-RAM model with word size $w$, we show that we can construct in $o(w)$ time an error correcting code with a constant relative positive distance that maps numbers of $w$ bits into $\Theta(w)$-bit numbers, and such that the application of the error-correcting code on any given number $x\in[0,2^w-1]$ takes constant time. Our result improves on a previously proposed error-correcting code with the same properties whose construction time was exponential in $w$.