Avinash Sathaye

Multiple error detection and identification via signature analysis

AbstractSignature analysis has been used widely for fault detection as a part of Built-In Self Test (BIST). In this paper we show how signature analysis can be used not only for fault detection but also for identification of multiple errors produced by faults in the circuits under test. We construct Signature Analysis Registers (SARs) to detect and identify any specified number of errors in the input polynomials by choosing proper characteristic polynomials. To detect and identifyr errors in an input bit stream ofm bits, we use a polynomialgr(x)=1cm (f1(x), f3(x), ..., f2r−1(x)) as the characteristic polynomial for the SAR for any polynomialf1(x), where lcm represents the least common multiple of polynomials al

On automorphisms and derivations of Cayley-Dickson algebras

fi(x) = Res_t (f_1 (t),x - t^i ), i = 3,...,2r - 1,

Applications of one-dimensional cellular automata and linear feedback shift registers for pseudo-exhaustive testing

Rest denotes thet-Resultant, andm is less than the order off1(x). Given a faulty signature produced by an SAR constructed as described, we present an algorithm for the identification of the actual error bits in the input polynomial to the SAR. We also extend the use of BCH codes for error detection and correction to include nonprimitive polynomials.

Uniqueness of plane embeddings of special curves

In case k is a field of characteristic zero, it is a conjecture that the answer to (2) is “yes” and the answer to (2) is known to be “no” if k has positive characteristic. No counterexamples to (1) are known when k is a field. If k is a field, special cases of (1) and (2) have been recently studied by the following method. Let ALlI = A[T]. F or the cancellation problem take F to be a suitable variable in C so that C = k[F][“l and F # A. Now identify A as a subring of A[TJ/(F) = B. Then one explicitly constructs variables for A in terms of judiciously chosen variables for B w kc21, exploiting the fact that B is a simple ring extension of A. In [9] and [7] th iswasdoneforF=bT+awitha,bEA and in [lo] the case F = bTn + a with n > 1 and coprime with the characteristic of k was treated. In [g], the ideas of [lo] have been extended to equations F such that B is Galois over A, i.e., A and BC have the same quotient field, where G = Aut, B.

Some remarks on the Jacobian question

where (y,) is a sequence of nonzero elements of F called the sequence of structure constants. Albert called these Cayley-Dickson algebras. The construction, extended to general initial algebras A,, is known as the Cayley-Dickson process [Sl, p. 451 and the algebras A,, that it generates are known as generalized Cayley-Dickson algebras of order n. In the “classical case”, A, is the real numbers and yi = -1 for each i. Then A, is the complex numbers, A1 the quaternions, and A, the Cayley numbers. We are concerned here with the case when A, is a field, F, of characteristic other than 2 or 3, its involution is the identity function, and {y,} is an arbitrary sequence of nonzero elements in F. For n B 3 these Cayley-Dickson algebras are not associative but they do satisfy the flexible law (x(yx) = (xy)x) for each pair x, y E A,, [S2]. This can be stated as saying that the associator (x, y, x) vanishes identically. The strong nonassociutiuity conjecture of P. Yiu [YIU] asserts that the flexible law is the only identity of this type. It says that for x, ZE A,, the associator (x, y, z) vanishes identically if and only if 1, x, and z are linearly dependent over F. We prove a special case of this conjecture (Proposition 1.5) which we then

Alternating group coverings of the affine line for characteristic two

(See w 2 for notations.) F-Conjecture when dim A =3 was proved in [M-l] for unmixed complete intersection p and the F-conjecture was also proved in [A] for dim A = 3 and also when Alp is a polynomial ring in the general case. Murthy generalized his results in [M-I] to include the cases of all prime ideals p of height ~ 2 which are locally complete intersections [M-2, Theorem 2.2]. We prove the following:

Geometric theory of algebraic space curves

Recent studies have shown that cellular automata (CA) can be used in place of linear feedback shift registers (LFSRs). Due to their regular structure and local feedback connections, VLSI layout of CA is easy and results in saving silicon area. Some properties of CAs derived using Rules 90 and 150 in terms of characteristic polynomials are presented. In particular, (a) computation of characteristic polynomials in CA, (b) computation of sequence lengths generated by various CA/LFSRs, and (c) pseudo-exhaustive testing by sequences generated by CA/LFSRs are accomplished. >

Translates of Polynomials

For a family of special affine plane curves, it is shown that their embeddings in the affine plane are unique up to automorphisms of the affine plane. Examples are also given for which the embedding is not unique. We also discuss the Lin-Zaidenberg estimate of the number of singular points of an irreducible curve in terms of its rank. Formulas concerning the rank of the curve lead to an alternate simpler version of the proof of the Epimorphism Theorem.