In this work, we prove new results concerning the combinatorial properties of random linear codes. Firstly, we prove a lower bound on the list-size required for random linear codes over $\mathbb F_q$ $\varepsilon$-close to capacity to list-recover with error radius $\rho$ and input lists of size $\ell$. We show that the list-size $L$ must be at least $\frac{\log_q\binom{q}{\ell}-R}{\varepsilon}$, where $R$ is the rate of the random linear code. As a comparison, we also pin down the list size of random codes which is $\frac{\log_q\binom{q}{\ell}}{\varepsilon}$. This leaves open the possibility (that we consider likely) that random linear codes perform better than random codes for list-recoverability, which is in contrast to a recent gap shown for the case of list-recovery from erasures (Guruswami et al., IEEE TIT 2021B). Next, we consider list-decoding with constant list-sizes. Specifically, we obtain new lower bounds on the rate required for list-of-$3$ decodability of random linear codes over $\mathbb F_2$; and list-of-$2$ decodability of random linear codes over $\mathbb F_q$ (for any $q$). This expands upon Guruswami et al. (IEEE TIT 2021A) which only studied list-of-$2$ decodability of random linear codes over $\mathbb F_2$. Further, in both cases we are able to show that the rate is larger than that which is possible for uniformly random codes.
翻译:在这项工作中, 我们证明随机线性代码的组合属性有新的结果。 首先, 我们证明, 美元是随机线性代码的速率, 其列表大小比 $\ mathbf F_ q$_ q_ qbinom{q_ qell_ varepsilon} 更接近于以错误半径$\ rho$ 美元和大小为$\ ell$的输入列表重新覆盖的能力。 我们显示, 列表大小的美元必须至少是$\\ log_ q\ binom{qhell{ qhrepsilon} $美元。 我们显示, 列表大小至少为$21 美元 qtreme2 $- rassuralwami 。 在随机线性代码中, 将 $20- etelwami et etal codeal 值比 $ $ 20_ tIT 2021B美元 。 的 Orality lecomblecable licolal lex lex ladeal ladeal ladeal dex code, 我们考虑以 codeal codeal codeal dromodeal dow_ frodudeal codeal downal d list downal_ fromodal_ frobal libledal ledal_ fol liblational libal libledal_ libal ledaldaldal ledal libaldals libal libs lications libal libal libal libal libal libal libal ledal libal libal lical libal libal lical libal libal le le lical lical lid le le le le le ledal led led led led ledal le le le le le libal le le