TY - GEN

T1 - Testing polynomials over general fields

AU - Kaufman, Tali

AU - Ron, Dana

PY - 2004

Y1 - 2004

N2 - In this work we fill in the knowledge gap concerning testing polynomials over finite fields. As previous works show, when the cardinality of the field, q, is sufficiently larger than the degree bound, d, then the number of queries sufficient for testing is polynomial or even linear in d. On the other hand, when q = 2 then the number of queries, both sufficient and necessary, grows exponentially with d. Here we study the intermediate case where 2 < q ≤ O(d) and show a smooth transition between the two extremes. Specifically, let p be the characteristic of the field (so that p is prime and q = ps for some integer s ≥ 1). Then the number of queries performed by the test grows like l · q2l+1 where l = [d+1/q-q/p]. Furthermore, q Ω(l) queries are necessary when q ≤ O(d). The test itself provides a unifying view of the two extremes: it considers random affine subspaces of dimension l and verifies that the function restricted to the selected subspaces is a degree d polynomial. Viewed in the context of coding theory, our result shows that Reed-Muller codes over general fields (usually referred to as Generalized Reed-Muller (GRM) codes) are locally testable. In the course of our analysis we provide a characterization of small-weight words that span the code. Such a characterization was previously known only when the field size is a prime or is sufficiently large, in which case the minimum weight words span the code.

AB - In this work we fill in the knowledge gap concerning testing polynomials over finite fields. As previous works show, when the cardinality of the field, q, is sufficiently larger than the degree bound, d, then the number of queries sufficient for testing is polynomial or even linear in d. On the other hand, when q = 2 then the number of queries, both sufficient and necessary, grows exponentially with d. Here we study the intermediate case where 2 < q ≤ O(d) and show a smooth transition between the two extremes. Specifically, let p be the characteristic of the field (so that p is prime and q = ps for some integer s ≥ 1). Then the number of queries performed by the test grows like l · q2l+1 where l = [d+1/q-q/p]. Furthermore, q Ω(l) queries are necessary when q ≤ O(d). The test itself provides a unifying view of the two extremes: it considers random affine subspaces of dimension l and verifies that the function restricted to the selected subspaces is a degree d polynomial. Viewed in the context of coding theory, our result shows that Reed-Muller codes over general fields (usually referred to as Generalized Reed-Muller (GRM) codes) are locally testable. In the course of our analysis we provide a characterization of small-weight words that span the code. Such a characterization was previously known only when the field size is a prime or is sufficiently large, in which case the minimum weight words span the code.

UR - http://www.scopus.com/inward/record.url?scp=17744373121&partnerID=8YFLogxK

U2 - 10.1109/focs.2004.65

DO - 10.1109/focs.2004.65

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AN - SCOPUS:17744373121

SN - 0769522289

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 413

EP - 422

BT - Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Sciences, FOCS 2004

T2 - Proceedings - 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004

Y2 - 17 October 2004 through 19 October 2004

ER -