2016
DOI: 10.1007/s10623-016-0240-5
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On the number of irreducible linear transformation shift registers

Abstract: We deal with the problem of counting the number of irreducible linear transformation shift registers (TSRs) over a finite field. In a recent paper, Ram reduced this problem to calculate the cardinality of some set of irreducible polynomials and got explicit formulae for the number of irreducible TSRs of order two. We find a bijection between Ram's set to another set of irreducible polynomials which is easier to count, and then give a conjecture about the number of irreducible TSRs of any order. We also get exp… Show more

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Cited by 4 publications
(6 citation statements)
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“…We believe that applying such equivalence and thus reducing to a set of simpler expressions provides a more systematic approach when tacking tasks such as the counting problem addressed in this paper. In support of this view, in the final Section 7 we use our results to swiftly recover formulas for the number of irreducible TSRs of order three over F q m , which were originally obtained by Jiang and Yang in [8].…”
Section: Introductionsupporting
confidence: 58%
See 1 more Smart Citation
“…We believe that applying such equivalence and thus reducing to a set of simpler expressions provides a more systematic approach when tacking tasks such as the counting problem addressed in this paper. In support of this view, in the final Section 7 we use our results to swiftly recover formulas for the number of irreducible TSRs of order three over F q m , which were originally obtained by Jiang and Yang in [8].…”
Section: Introductionsupporting
confidence: 58%
“…This connection, together with Ahmadi's evaluation, in [1], of |I(g(x), h(x), m, q)| for the case where g(x)/h(x) is quadratic, was then used in [4,Theorem 3.4] to obtain a new derivation of an explicit formula for | TSRI(m, 2, q)|, originally found by Ram [15] by a different method. We use a similar method to recover the formulas for | TSRI(m, 3, q)| obtained by Jiang and Yang in [8].…”
Section: Transformation Shift Registersmentioning
confidence: 99%
“…As mentioned in [JY17], the results of Theorem 22 can be expressed in terms of counting polynomials of various types if one so wishes, akin to what we have done in previous sections. The most pleasant expression is obtained when q ≡ 1 (mod 3), where we find | TSRI(m, 3, q)| = |GL m (F q )| q m − 1 • q(q + 1) 3m 3 I(m/m 3 , q m 3 ) − I(m/m 3 , 1) .…”
Section: Transformation Shift Registersmentioning
confidence: 92%
“…This connection, together with Ahmadi's evaluation, in [Ahm11], of |I(g(x), h(x), m, q)| for the case where g(x)/h(x) is quadratic, was then used in [CHPW15,Theorem 3.4] to obtain a new derivation of an explicit formula for | TSRI(m, 2, q)|, originally found by Ram [Ram15] by a different method. We use a similar method to recover the formulas for | TSRI(m, 3, q)| obtained by Jiang and Yang in [JY17].…”
Section: Transformation Shift Registersmentioning
confidence: 99%
See 1 more Smart Citation