On the covering radius of some binary cyclic codes

We compute the covering radius of some families of binary cyclic codes. In particular, we compute the covering radius of cyclic codes with two zeros and minimum distance greater than 3. We also compute the covering radius of some binary primitive BCH codes over \begin{document}$\mathbb{F}_{2^f}$\end{document} , where \begin{document}$f=7, 8$\end{document} .


Introduction
The covering radius of a code is one of the important parameters in the theory of error correcting codes. Codes with a 'good' covering radius have applications in the theory of communications, e.g. data compression, decoding of errors and erasures ( [3,7,11]).
Let C be a code over F 2 f of length n. The covering radius of a code C is the smallest ρ such that the spheres B ρ (c) = {v ∈ F n 2 f | d(v, c) ≤ ρ} cover F n 2 f , where d(v, c) is the Hamming distance between v and c ∈ C, i.e. ρ = max The calculation of covering radius of cyclic codes is an active area of research in applied mathematics. The covering radius of the code C 2 i +1,1 with zeros α 2 i +1 and α is known for gcd(i, f ) = 1 (this is a quasi-perfect code). The code C 2 i +1,1 over F 2 f has covering 3 since the dual of the code C 2 i +1,1 has three nonzero weights (see [3,7,12,13]). In [15], the authors proved that the covering radius of C 2 i +1,1 over F 2 f is 3 when f is even. Quasi-perfect codes with covering radius 3 have received special attention ( [8-10, 14-16, 21]).
In [15][16][17], the authors introduced a method to obtain the covering radius of binary primitive cyclic codes. They used the Moreno-Moreno's theorem ( [18,19]) to prove the solvability of systems of two or three equations. In this paper, we improve Moreno-Moreno's theorem for systems of two diagonal equations. Using this improvement we compute the covering radius of codes that are not included in [15][16][17]. Also, we modify the method introduced in [15][16][17] to compute the covering radius of some cyclic codes over F 2 f , where f ≤ 13 and f is odd. Finally, we prove that the method introduced in [15][16][17] can be applied to codes with smaller minimum distance. Finally, we discuss an algorithm that allowed us to compute the covering radius of some BCH codes for finite fields F 128 and F 256 .

Preliminaries
In this section we discuss all the necessary tools that will be used in rest of the sections.
1 · · · X eni n be a nonconstant polynomial over a finite fied F 2 f . Let ν 2 (a) be the 2-valuation of the integer a.
Let φ : F 2 f → Q be a nontrivial additive character. The exponential sum associated to F (X) is defined as follows: The next theorem gives a lower bound for the 2-valuation of an exponential sum.

Divisibility of diagonal equations
In this section we improve Moreno-Moreno's result for systems of two diagonal equations. This improvement is a key ingredient in our calculations of the covering radius of cyclic codes with two zeros.
Multiplying the first four modular equations of (4) by D, we obtain Now we apply σ 2 to system (5) to get Hence

Applications
In this section we will compute the covering radius of some binary cyclic codes over F 2 f . Let α be a primitive element of F 2 f and let C d1,d2,...,dt be the code with zeros α d1 , . . . , α dt over F 2 f . Lemma 4.1. Let α be a primitive element of F 2 f .
1. Let C d1,d2 be the code with zeros α d1 , α d2 over F 2 f . If the minimum distance of C d1,d2 is greater than 3, then the number of solutions N d1,d2,f of (8) over We are going to prove the first part of the lemma and give a brief outline of the proof of the second part. Using the fact that the minimum distance of C d1,d2 is greater than 3, the system (8) has only trivial solutions. Let Let A i = {(x 1 , . . . , x 5 ) | x i = 0, (x 1 , . . . , x 5 ) be a solution of (9)} for i = 1, . . . , 5. Then the number of solutions of system (9) is equal to N d1,d2,d3,f = |A 1 ∪ . . . ∪ A 5 |.

Using the principle of inclusion-exclusion with
Theorem 4.2. Let α be a primitive element of F 2 f with f odd. Let C d1,d2 be the code with zeros α d1 , α d2 over F 2 f . If the minimum distance of C d1,d2 is greater than 3, then the covering radius of C d1,d2 is 3, whenever the sequence d 1 , d 2 is 2-adic bounded by 2.
Proof. We need to prove that the following system of equations has at least one solution for any (α 1 , α 2 ) ∈ F 2 2 f (see [3,7,11]) . Now we consider the homogenization of system (10): (10) does not have a solution for some (α 1 , α 2 ), then the number of solutions of (11) is equal to number of solutions of the following system We are going to prove that the number of solutions of (11) cannot equal to the number of solutions of (12). Hence the system (11) has a solution (x 1 , x 2 , x 3 , x 4 ) with x 4 = 0. Therefore ( x1 x4 , x2 x4 , x3 x4 ) is a solution of (10). Suppose there exists (α 1 , α 2 ) such that the system (10) does not have a solution. Then systems (11) and (12) have the same number of solutions. We have that the number of solutions of system (12) is equal to 3·2 f −2 by Lemma 4.1 and 4 divides the number of solutions of system (11) by Theorem 3.1. This is a contradiction since 4 does not divide 3 · 2 f − 2. Hence system (10) has at least one solution for any (α 1 , α 2 ) ∈ F 2 2 f . The following corollary is an immediate implication of Theorem 4.2.
• Let k 1 , k 2 , f be positive integers with k 1 = k 2 and f is odd. If the minimum distance C 2 k 2 +1,2 k 1 +1 is greater than 3, then the covering radius of C 2 k 2 +1,2 k 1 +1 is 3.
Then the corollary follows applying Theorem 4.2.
• The code C 1,13 has covering radius 3, whenever f is odd.
Proof. In [6], the authors proved that those codes have minimum distance > 3. Applying Theorem 4.2 follows Corollary 3.

Remark 2.
When the minimum distance of the code in previous results is 5 we obtain a quasi-perfect code. In [5], Moreno et al. stated the following conjecture: Prove that all the binary primitive cyclic codes with two zeros and minimum distance 5 are quasi-perfect.
Theorem 4.3. Let α be a primitive element of F 2 f with f odd. Let C d1,d2,d3 be the code with zeros α d1 , α d2 , α d3 over F 2 f . If the minimum distance of C d1,d2,d3 is greater than 5, then the covering radius of C d1,d2,d3 is 5, whenever 32 divides the number of solutions of the following system: Proof. In this case we need to prove that the following system is solvable for any (α 1 , α 2 , α 2 ) (see [3,7,11]) x d3 1 + x d3 2 + · · · + x d3 5 = α 3 . Suppose there exists (α 1 , α 2 , α 3 ) such that system (14) does not have a solution. Then the number of solutions of systems (9) and (13) are equal. From Lemma 4.1, the number of solutions of (9) over F 2 f is 15·2 f (2 f −2)+16. Using Moreno-Moreno's theorem 2 f /2 divides the number of solutions of (13). This is a contradiction since 32 does not divide the number solutions of (9) for f > 8 and 32 divides the number solutions of (13) .
Note that Theorem 4.3 does not require f to be odd.
In the next section we use a computer program to calculate the covering radius of some codes of the type C 1,2 i +1,2 2i +1 for f ≤ 8.

Calculations
In this section we present the results of computing the covering radius of cyclic codes. In Tables 1 and 2 we combine the calculation of the minimum of a modular equation and Theorem 4.2 to compute the covering radius of cyclic codes of the type C 1,d over F 2 f with f = 7, 9, 11, 13. In Tables 3 and 4 we compute the covering  radius of some BCH codes adding values to the Table 10.1 in [7] and the covering radius of codes not included in Corollary 4.
We use a bit representation for the elements of field F 2 f , i.e. each element is represented by a word of f bits where the bit in position i represents the coefficient of monomial degree i. For example, element β 3 + β 2 + 1 in F 2 7 is represented as 0001101. To generate a solution for α 1 = k we take advantage of the fact that its bits are the result from bitwise XORing (x 1 , . . . , x n ). Figure 1 illustrates a solution for α 1 = 1 ∈ F 2 7 . Looking at each bit position (the columns) we notice that to generate a given solution we must provide a combination of column values such that they have an even/odd number of 1's to produce a 0/1 in their bit position. In the Figure 1, all columns except the least significant require an even number of 1's. We construct solutions for a given α 1 by exhaustively generating all combinations of bit columns with even/odd number of 1's at the Algorithm 1: Algorithm to determine if all (α 2 , . . . , α n ) ∈ F 2 f exist for α 1 = k Data: k ∈ F 2 f Result: returns true if all (α 2 , . . . , α t ) ∈ F 2 f exist for α 1 = k C is an array of size 2 f (t−1) initialized to 0's; ctr ← 0; while next solution exists(k) AND ctr < 2 f (t−1) do (x 1 , . . . , x n ) ← next solution(k); compute α 2 , . . . , α t for (x 1 , . . . , x m ); if C[α 2 , . . . , α t ] == 0 then C[α 2 , . . . , α t ] ← 1; ctr ← ctr + 1; end end if ctr == 2 f (t−1) then return true; else return false; end proper positions, then extracting the values for the (x 1 , . . . , x n ). A C-language implementation of the algorithm can be downloaded at: https://bitbucket.org/ raarceupr/reed-francis. = β 3 + β = β 5 + β 3 + β 2 + β + 1 = β 6 + β 3 + β 2 = β 6 + β 5 + β 3 + 1 Figure 1. Construction of a solution for α 1 = 1 ∈ F 2 7 . By choosing numbers that have an even quantity of 1's for all columns except the least significant, we guarantee that solution α 1 = 0000001. The values for (x 1 , . . . , x 5 ) are read from the rows. Table 3 contains the covering radius of binary primitive BCH codes not included in Table 10.1 in [7]. Table 4 contains the covering radius of cyclic codes of the type C 1,2 i +1,2 2i +1 over F 2 f that are not included in Corollary 4.