Lifted projective Reed–Solomon codes

Abstract

Lifted Reed–Solomon codes, introduced by Guo et al. (in: Kleinberg (ed) Proceedings of the 4th conference on innovations in theoretical computer science, ITCS’13, ACM, New York, 2013), are known as one of the few families of high-rate locally correctable codes. They are built through the evaluation over the affine space of multivariate polynomials whose restriction along any affine line can be interpolated as a low degree univariate polynomial. In this work, we give a formal definition of their analogues over projective spaces, and we study some of their parameters and features. Local correcting algorithms are first derived from the very nature of these codes, generalizing the well-known local correcting algorithms for Reed–Muller codes. We also prove that the lifting of both Reed–Solomon and projective Reed–Solomon codes are deeply linked through shortening and puncturing operations. It leads to recursive formulae on their dimension and their monomial bases. We finally emphasize the practicality of lifted projective Reed–Solomon codes by computing their information sets and by providing an implementation of the codes and their local correcting algorithms.

Appendices

Appendix A: A useful results

1.1 A.1 Combinatorial Nullstellensatz

We recall the Combinatorial Nullstellensatz proved by Alon in [1].

Theorem 7

(Combinatorial Nullstellensatz [1]) Let $\mathbb {F}$ be a field and $f \in \mathbb {F}[X_1, \dots , X_r]$. Assume that $\deg (f) = \sum _{i=1}^r t_i$ and the coefficient of the monomial $\mathbf {X^t} = \prod _{i=1}^r X_i^{t_i}$ in f is non-zero (in other words, assume that $\mathbf {t} = (t_1, \dots , t_r)$ is a degree of f). Let finally $W_1, \dots , W_r \subseteq \mathbb {F}$ such that $|W_i| > t_i$ for every $1 \le i \le r$.

Then, there exists $\mathbf {w} \in W_1 \times \dots \times W_r$ such that $f(\mathbf {w}) \ne 0$.

1.2 A.2 Technical results

Lemma 10

The following equality over bivariate polynomials holds:

$$\begin{aligned} \sum _{\beta \in \mathbb {F}_q} (\beta X + Y)^{q-1} = -X^{q-1}\,. \end{aligned}$$

Proof

Let $\alpha$ be a primitive element of $\mathbb {F}_q$.

$$\begin{aligned} \begin{aligned} \sum _{\beta \in \mathbb {F}_q} (\beta X + Y)^{q-1}&= Y^{q-1} + \sum _{i=0}^{q-2} (\alpha ^i X + Y)^{q-1} = Y^{q-1} + \sum _{i=0}^{q-2} \sum _{j=0}^{q-1} \left( {\begin{array}{c}q-1\\ j\end{array}}\right) \alpha ^{ij} X^j Y^{q-1-j}\\&= Y^{q-1} + \sum _{j=0}^{q-1} \left( {\begin{array}{c}q-1\\ j\end{array}}\right) \left( \sum _{i=0}^{q-2} (\alpha ^j)^i \right) X^j Y^{q-1-j}\\&= Y^{q-1} + \left( \left( {\begin{array}{c}q-1\\ 0\end{array}}\right) \times (-1) \times X^0 Y^{q-1} \right) + \left( \left( {\begin{array}{c}q-1\\ q-1\end{array}}\right) \times (-1) \times X^{q-1} Y^0 \right) \\&= -X^{q-1} \end{aligned} \end{aligned}$$

1.3 A.3 Automorphism groups of (projective) Reed–Muller codes

The automorphism group of affine Reed–Muller codes has been thoroughly studied by Berger and Charpin in [4] with group algebra techniques. For our needs, we recall below that this group contains the subgroup of affine transformations.

Proposition 8

(Reed–Muller code) Let $0 \le k \le m(q-1)$. The automorphism group of the Reed–Muller code $\mathcal {C}= \mathrm {RM}_q(m, k)$ contains the affine permutations $\mathrm {Aff}(\mathbb {F}_q, m)$.

Proof

Let $c = {{\mathrm{ev}}}_{\mathbb {A}^m}(f) \in \mathcal {C}$, $M \in \mathrm {GL}_m(\mathbb {F}_q)$ and $\mathbf {b} \in \mathbb {F}_q^m$. Denote by $T_{M, \mathbf {b}}(\mathbf {x}) = M\mathbf {x} + \mathbf {b}$ for every $\mathbf {x} \in \mathbb {A}^m$. Let us prove that $T_{M, \mathbf {b}}(c) \in \mathcal {C}$.

We remark that $T_{M, \mathbf {b}}(c) = {{\mathrm{ev}}}_{\mathbb {A}^m}(f \circ T_{M, \mathbf {b}}^{-1})$. Since $T_{M, \mathbf {b}}$ is affine, so is $T_{M, \mathbf {b}}^{-1}$, and the total degree of $f \circ T_{M, \mathbf {b}}^{-1}$ is the same that the total degree of f. Hence ${{\mathrm{ev}}}_{\mathbb {A}^m}(f \circ T_{M, \mathbf {b}}^{-1}) \in \mathrm {RM}_q(m, k)$ and the proof is completed.

A few years later, Berger also studied the automorphism group of projective Reed–Muller codes [3].

Proposition 9

(Projective Reed–Muller code) Let $0 \le v \le m(q-1)$. The automorphism group of the projective Reed–Muller code $\mathcal {C}= \mathrm {PRM}_q(m, v)$ contains the projective automorphisms $\mathrm {Proj}(\mathbb {F}_q, m)$.

Proof

Using that

$$\begin{aligned} {{\mathrm{ev}}}_{\mathbb {P}^m}(f \circ M) = w_M^v \,\star \, \sigma _{M^{-1}}({{\mathrm{ev}}}_{\mathbb {P}^m}(f)) \end{aligned}$$

for every $(w_M^v, \sigma _{M^{-1}}) \in \mathrm {Proj}(\mathbb {F}_q, m)$, the proof is very similar to the previous one.

Appendix B: Building the query generator $\mathcal {R}_s$

We recall that in our local correction algorithm (Sect. 4.2) we need a randomized query generator $\mathcal {R}_s$ which, given a point $\mathbf {P} \in \mathbb {P}^m$ and an embedding $L \in \mathrm {Emb}_\mathbb {P}(m, \mathbf {P})$, returns s random points of $L(\mathbb {P}^1)$ such that:

$$\begin{aligned} \forall \mathbf {Q} \in \mathbb {P}^m, \mathrm {Pr}_{\begin{array}{c} L \leftarrow \mathrm {Emb}_\mathbb {P}(m, \mathbf {P})\\ S \leftarrow \mathcal {R}_s(\mathbf {P}, L) \end{array}} [ \mathbf {Q} \in S ] = s/n\,. \end{aligned}$$

(8)

The tricky point is that, for a fixed $L \in \mathrm {Emb}_\mathbb {P}(m, \mathbf {P})$, we cannot pick the s points uniformly at random on $L(\mathbb {P}^1)$, otherwise the point $\mathbf {P}$ will have a larger probability to be chosen than the other points. We provide a solution to this issue in Algorithm 2.

Appendix C: Computation of the dimension of lifted codes

In the following tables are presented some parameters of affine lifted codes, projective lifted codes and projective Reed–Muller codes. We denote respectively by $n_A$, $\dim (A)$ and $R_A$ the length, dimension and rate of $A = \mathrm {Lift}_q(m, k-1)$ the value of k given in the first row (q and m being fixed in each table). Similarly, $n_P$, $\dim (P)$ and $R_P$ represent the length, dimension and rate of $P = \mathrm {PLift}_q(m, k)$, while $\dim (\mathrm {PRM})$ and $R_{\mathrm {PRM}}$ denote the dimension and the rate of $\mathrm {PRM}_q(m, k)$ (its length being $n_P$).

We choose to compare these codes because, in the local correcting algorithm, they admit approximately the same error-correction capability and locality. Our goal is to show how lifting leads to higher rates, and that projective and affine lifted codes behave similarly.

1.1 C.1 Parameters of the kind $m=2$, $q=2^t$

See Tables 2, 3, 4, 5 and 6.

Table 2 Parameters for $m=2$ and $q=4$

Table 3 Parameters for $m=2$ and $q=8$

Table 4 Parameters for $m=2$ and $q=16$

Table 5 Parameters for $m=2$ and $q=32$

Table 6 Parameters for $m=2$ and $q=64$

1.2 C.2 Parameters of the kind $m=3$, $q=2^t$

See Tables 7, 8, 9, 10 and 11.

Table 7 Parameters for $m=3$ and $q=4$

Table 8 Parameters for $m=3$ and $q=8$

Table 9 Parameters for $m=3$ and $q=16$

Table 10 Parameters for $m=3$ and $q=32$

Table 11 Parameters for $m=3$ and $q=64$