New Constructions of Optimal Locally Recoverable Codes via Good Polynomials

Constructions of Locally Recoverable Codes Which are Optimal

“…The following result is [7, Construction 1]. We write it in the format of [3,Theorem 4], which is more convenient for our purposes.…”

“…The strenght of our method relies on the fact that we do not need to base our constructions on algebraic properties of the base field as in [3,7] but we extract good polynomials with a density argument. Let us now show with a toy example this fact.…”

“…Let us now show with a toy example this fact. Suppose that we want to construct via good polynomials a (n, k, 2)-LCR code over an alphabet of size q, with q ≡ 2 mod 3 using one of the constructions in [3]. One would need a degree 3 polynomial that is totally split at at least some place t 0 ∈ F q .…”

“…But then this cannot be a composition, as its degree is prime and cannot be a linearised polynomial, as its degree is incompatible with the characteristic. Also, it cannot be a power because x → x 3 is a bijection by the congruence class of q modulo 3, so the constructions in [3] do not apply. Of course, one could get an (n, k, 2)-LRC code by extending the base field to F q 2 (instead of F q ) and using the good polynomial x 3 .…”

Codes With Hierarchical Locality From Covering Maps of Curves

“…Take n " 36, ν " 12, r 2 " 3, r 1 " 6, t " 2, then dimpCq " 12 and d min " 18 from Eq. (14), and thus the code C has parameters [36, 12,18] and is distance-optimal. The code is constructed as follows.…”