On deep holes of generalized Reed-Solomon codes

Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word $u$ is a deep hole of the standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. In this paper, we extend this result by giving a new class of deep holes of the generalized Reed-Solomon codes.


Introduction and the statement of the main result
Let F q be the finite field of q elements with characteristic p.Let n and k be positive integers.Let D = {x 1 , ..., x n } be a subset of F q , which is called the evaluation set.The generalized Reed-Solomon code C q (D, k) of length n and dimension k over F q is defined as follows: q , then it is called standard Reed-Solomon code.If D = F q , then it is called extended Reed-Solomon code.For any [n, k] q linear code C, the minimum distance d(C) is defined by d(C) := min{d(x, y)|x ∈ C, y ∈ C, x = y}, where d(•, •) denotes the Hamming distance of two words which is the number of different entries of them and w(•) denotes the Hamming weight of a word which is the number of its nonzero entries.Thus we have The error distance to code C of a received word u ∈ F n q is defined by The most important algorithmic problem in coding theory is the maximum likelihood decoding (MLD): Given a received word, find a word v ∈ C such that d(u, v) = d(u, C) [5].Therefore, it is very crucial to decide d(u, C) for the word u.Sudan [6] and Guruswami-Sudan [2] provided a polynomial time list decoding algorithm for the decoding of u when d(u, C) ≤ n − √ nk.When the error distance increases, the decoding becomes NP-complete for the generalized Reed-Solomon codes [3].
When decoding the generalized Reed-Solomon code C, for a received word u = (u 1 , ..., u n ) ∈ F n q , we define the Lagrange interpolation polynomial u(x) of u by i.e., u(x) is the unique polynomial of degree at most n − 1 such that u(x i ) = u i for 1 ≤ i ≤ n.For u ∈ F n q , we define the degree of u(x) to be the degree of u, i.e., deg(u Evidently, we have the following simple bounds. then the upper bound is equal to the lower bound, and so d(u, C) = n − k which implies that u is a deep hole.This gives immediately (q − 1)q k deep holes.We call these deep holes the trivial deep holes.It is an interesting open problem to determine all deep holes.Cheng and Murray [1] showed that for the standard Reed-Solomon code [p − 1, k] p with k < p 1/4−ǫ , the received vector (f (α)) α∈F * p cannot be a deep hole if f (x) is a polynomial of degree k + d for 1 ≤ d < p 3/13−ǫ .Based on this result, they conjectured that there is no other deep holes except the trivial ones mentioned above.Li and Wan [5] used the method of character sums to obtain a bound on the non-existence of deep holes for the extended Reed-Solomon code C q (F q , k).Wu and Hong [8] found a counterexample to the Cheng-Murray conjecture [1] about the standard Reed-Solomon codes.
Let l be a positive integer.In this paper, we investigate the deep holes of the generalized Reed-Solomon codes with the evaluation set D := F q \ {a 1 , ..., a l }, where a 1 , ..., a l are any fixed l distinct elements of F q .Our method here is different from that of [8].Write D = {x 1 , ..., x q−l } and for any f (x) ∈ F q [x], let f (D) := (f (x 1 ), ..., f (x q−l )).
Then we can rewrite the generalized Reed-Solomon code C q (D, k) with evaluation set D as Actually, by constructing some suitable auxiliary polynomials, we find a new class of deep holes for the generalized Reed-Solomon codes.That is, we have the following result.
Theorem 1.2.Let q ≥ 4 and 2 ≤ k ≤ q − l − 1.For 1 ≤ j ≤ l, we define where λ j ∈ F * q and r j (x) ∈ F q [x] is a polynomial of degree at most k−1.Then the received words u 1 (D), ..., u l (D) are deep holes of the generalized Reed-Solomon code C q (D, k).
The proof of Theorem 1.2 will be given in Section 2. The materials presented here form part of the second author's PhD thesis [7], which was finished on April 15, 2012.

Proof of Theorem 1.2
Evidently, for any a ∈ F q , we have (a − a j ) = a q − a = 0, and for any a ∈ D, we have N (a) = 0, where we denote the reduction of f (x) mod N (x).Therefore, for any x i ∈ D, we have f First of all, we give a lemma about error distance.In what follows, we let G k denote the set of all the polynomials in F q [x] of degree at most k − 1.
Lemma 2.1.Let #(D) = n and let u, v ∈ F n q be two words.
Proof.From the definition of error distance and noting that f ≤k−1 (x) ∈ G k , we get immediately that as one desires.So Lemma 2.1 is proved.Now we are in the position to prove Theorem 1.2.
Then by (2.1), we infer that For any integer j with 1 ≤ j ≤ l, we let For any y ∈ D, we have y − a j = 0, and so f j (y) = 1 y−aj .We claim that (2.3) max In order to prove this claim, we pick k distinct nonzero elements c j1 , ..., c j k of F q \ {a t − a j } l t=1 (since k ≤ q − l − 1).Now we introduce the auxiliary polynomial g j (x) as follows: Then deg(g j (x)) = k − 1, and so g j (x) ∈ G k .Since for any y ∈ D, we have It then follows that c j1 + a j , ..., c j k + a j are the all roots of f j (x) − g j (x − a j ) = 0 over F q .Noticing that c j1 , ..., c j k ∈ F q \ {a 1 − a j , ..., a l − a j }, we have c j1 + a j , ..., c j k + a j ∈ D. Also D ⊆ F q .Therefore c j1 + a j , ..., c j k + a j are the all roots of f j (x) − g j (x − a j ) = 0 over D. Hence On the other hand, for any h(x) ∈ G k , the equation 1 − (x − a j )h(x) = 0 has at most k roots over F q , and so it has at most k roots over D. But From (2.4) and (2.5), we arrive at the desired result (2.3).The claim (2.3) is proved.Now from (2.2) and (2.3), we derive immediately that d(f j (D), C q (D, k)) = q − l − k.
In other words, f j (D) is a deep hole of the generalized Reed-Solomon C q (D, k).Finally, from (1.1) one can deduce that (2.6) u j (D) = λ j f j (D) + r j (D).
Since deg r j (x) ≤ k − 1, it then follows from (2.6) and Lemma 2.1 that u j (D) is a deep hole of C q (D, k) as required.This completes the proof of Theorem 1.2.✷