For each of the 15,314 annotated proteins, we partitioned coding-region SNPs into non-synonymous and synonymous, which permits pN/pS to be calculated for proteins with at least one synonymous SNP (see Additional file 1: Table S4A for a list of genes with pN/ pS ≥ 2 and Additional file 1: Table S4B for the results of a GO analysis).
Hence, for all r<9, the concatenated codes have a higher rate than the partition codes.
The partition codes as described above only aim at maximizing the rate of the code.
In that case, we claim that even the simple partition codes of Section 4.1 are asymptotically optimal.
The concatenated codes described in Section 5 enable (r,ℓ -cooperative locality with better rate and minimum distance as compared to those of partition codes.
Note that the partition codes presented here are special cases of codes with ((frac {r}{ell }, delta = ell + 1))-locality as studied in [9, 10] (cf. Section 2.2).
Here, note that it was shown in [33] that for a random erasure channel, the partition codes are asymptotically optimal in terms of achieving capacity.
We obtain better rate as compared to that of the partition codes (cf. Section 4.1), iff frac{3r}{4(r + 8)} > frac{r}{r + 16} Rightarrow~~r < 16. (15).
In the above construction of the partition codes, we use an (left (frac {r}{ell } + ell, frac {r}{ell }right)) MDS code to encode disjoint groups of (frac {r}{ell }) message symbols.
The rate (Rleft (frac {ell }{2}right) ) in (17) is strictly greater than the rate of the partition codes (frac {r}{r + ell ^{2}}) as long as (r < frac {ell ^{3}}{4}).
Indeed, it is possible to attain a rate of (frac {r}{r + ell ^{1+epsilon }}) for some ε > 0. Comparing with (7), we see that partition codes are near-optimal in this case.
