An Improved Bound on the Zero-Error List-Decoding Capacity of the 4/3 Channel

We prove a new upper bound on the size of codes <inline-formula> <tex-math notation="LaTeX">$C \subseteq \{1,2,3,4\}^{n}$ </tex-math></inline-formula> with the property that every four distinct codewords in <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> have a coordinate where they all differ. Specifically, we provide a self-contained proof that such codes have size at most <inline-formula> <tex-math notation="LaTeX">$2^{6n/19 + o(n)}$ </tex-math></inline-formula>, that is, rate bounded asymptotically by 6/19 ≤ 0.3158 (measured in bits). This improves the previous best upper bound of 0.3512 due to (Arikan 1994), which in turn improved the 0.375 bound that followed from general bounds for perfect hashing due to (Fredman and Komlós, 1984) and (Körner and Marton, 1988). Finally, using a combination of our approach with a simple idea which exploits powerful bounds on the minimum distance of codes in the Hamming space, we further improve the upper bound to 0.31477.

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