Guillaume Quintin's research while affiliated with University of Limoges and other places

Publications (10)

Article
Full-text available
Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer operations still require dedicated implementations. In this article, we survey existing algorithms for modular int...
Article
This article is devoted to algorithms for computing all the roots of a univariate polynomial with coefficients in a complete commutative Noetherian unramified regular local domain, which are given to a fixed common finite precision. We study the cost of our algorithms, discuss their practical performances, and apply our results to the Guruswami and...
Article
Full-text available
In this paper, we study generalized Reed-Solomon codes (GRS codes) over commutative and noncommutative rings, we show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context, and we investigate their complexities. Under some hypothesis, the study of noncommutative GRS codes over finite rings leads to th...
Article
Cylindrical Algebraic Decomposition (CAD, first introduced in [Col75]) of Euclidean space has become an important tool in mathematics and allows for practical quantifier elimination (QE) over the reals. Much research has gone into improving the projection ...
Article
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In this paper we investigate the structure of quasi-BCH codes. In the first part of this paper we show that quasi-BCH codes can be derived from Reed-Solomon codes over square matrices extending the known relation about classical BCH and Reed-Solomon codes. This allows us to adapt the Welch-Berlekamp algorithm to quasi-BCH codes. In the second part...
Article
This thesis studies the algorithmic techniques of list decoding, first proposed by Guruswami and Sudan in 1998, in the context of Reed-Solomon codes over finite rings. Two approaches are considered. First we adapt the Guruswami-Sudan (GS) list decoding algorithm to generalized Reed-Solomon (GRS) codes over finite rings with identity. We study in de...
Article
In this paper we design a decoding algorithm based on a lifting decoding scheme. This leads to a unique decoding algorithm for Reed-Solomon codes over Galois rings with a very low complexity, and a list decoding algorithm. We show that, using erasures in our algorithms, allows to decode more errors than half the minimum distance with a high probabi...
Conference Paper
Full-text available
Playout delay or buffering are commonly used in the case of streaming multimedia to ensure smooth playout. A large delay, however, is required for promising a high quality in display. Such significant delays consume huge on-chip memory. We show that when the constraints on output are slightly relaxed, the playout delay needed can be reduced to a ne...
Article
In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length m and ideals of M_l(Fq)[X]/(X^m-1). This permits to construct new classes of codes, namely qua...
Article
We present an algorithm for list decoding codewords of algebraic number field codes in polynomial time. This is the first explicit procedure for decoding number field codes whose construction were previously described by Lenstra and Guruswami. We rely on an equivalent of the LLL reduction algorithm for $\OK$-modules due to Fieker and Stehl\'e and o...

Citations

... Apart from this contribution, Intel HEXL utilizes several existing algorithms from previous work. The Mathemagix library [13] provides Intel AVX-accelerated implementations of modular integer arithmetic using a SIMD programming model. NFLlib [1] provides similar acceleration of primitives common to the ring Z /( + 1) using Intel SSE and Intel AVX2 instructions. ...
... Several research-decoding algorithms of RS codes have been proposed, such as in [99], which uses rings of matrices. In DVB, a decoding procedure has been presented for the RS (204,188); some parts of decoding can be carried out by shift registers. ...
... If A is a commutative ring, if f ∈ A[x] and if a ∈ A, then f (x + a) can be computed withÕ(deg f ) operations in A in a usual "divide and conquer" fashion; see for instance [6,Lemma 7] ...
... The designed divider is also able to handle the special case in which the most significant symbol of the syndrome is zero. Errors positions and values evaluation has been carried out with well-established Chien [25] and Forney [15,26] methods respectively. The whole designed architecture is, also in this case, pipe-lined. ...
... Recently, stochastic network calculus 1 [10] based approaches have been proposed for performance analysis of multimedia systems [18], [22]. These approaches, however, used the probabilistic calculus only partially: the input stream objects of a multimedia stream (e.g., frames) and their execution time are assumed to be deterministic. ...
... The dimension of C is k = 5 and its minimum distance is d min = 5. It follows from Theorem 8 that C ⊥ is 1, θ 5 , θ 5 -MT of block lengths (3,4,4) and dimension 6. Theorem 13 provides a GPM for C ⊥ whose Hermite normal form is ...
... List decoding was introduced separately by Elias in [2] and Wezencraft in [7]. Biasse and Quintin proposed an algorithm for list decoding of number field (N F ) codes [1]. Construction of codes using maximal order number fields was presented by Mair and Oggier in [4]. ...