Narrow Results

We eliminate 39 infinite families of possible principal graphs as part of the classification of subfactors up to index 5. A number-theoretic result of Calegari–Morrison–Snyder, generalizing Asaeda–Yasuda, reduces each infinite family to a finite number of cases. We provide algorithms for computing the effective constants that are required for this result, and we obtain 28 possible principal graphs. The Ostrik d-number test and an algebraic integer test reduce this list to seven graphs in the index range (4,5) which actually occur as principal graphs.

Let p be a prime ≡ 1 (mod 7). In this paper, we obtain an explicit expression for a primitive seventh root of unity (mod p) in terms of coefficients of a Jacobi sum of order 7 and also in terms of a solution of a Diophantine system of Leonard and Williams, and then obtain Euler's criterion for septic nonresidues D (mod p) in terms of this seventh root. Explicit results are given for septic nonresidues for D = 2, 3, 5, 7.

Generalized cyclotomic numbers of order $e\ge 1$ with respect to an odd prime power are obtained. Hence, explicit expressions for primitive idempotents in the ring $Z_4[x]/(x^{p^n}-1)$ are obtained in two cases, when the multiplicative order of 2 modulo $p^n$ is $\varphi (p^n)/2$ and $\varphi (p^n)/4$, where $p$ is an odd prime. Orthogonality and self-duality of some $Z_4$ cyclic codes are also discussed. Further, a method for obtaining cyclic self-dual/isodual codes of length $p^n$ over $Z_4$ is given.

Cyclotomic classes of order 2 with respect to a product of two distinct odd primes $p$ and $q$ are represented in some specific forms and using these forms an alternate proof of Theorem 3 of [C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl.4 (1998) 140–166] is given, when $n=pq$. Further, it is observed that these classes are related to $l$-cyclotomic cosets, where $O_p(l)=\frac{\varphi (p)}{2}$ and $O_q(l)=\varphi (q)$ such that gcd($\frac{\varphi (p)}{2},\varphi (q))=1$. Finally, arithmetic properties of some families in $Z[x]/〈x^{pq}-1〉$ and hence in $F_l[x]/〈x^{pq}-1〉$ are studied.

Arithmetic properties of some families in $\frac{\mathbb{Z}[x]}{〈x^{p^{t}q}-1〉}$ and hence in $\frac{{𝔽}_l[x]}{〈x^{p^{t}q}-1〉}$ are obtained by using the cyclotomic classes of order 2 with respect to $n=p^{t}q$, where $l$ is primitive root modulo $q$, $O_{p^t}(l)=\frac{\varphi (p^t)}{2}$ and $\mathrm{\text{gcd}}(\varphi (p^t),\varphi (q))=2$. The form of these cyclotomic classes enables us to generalize the results obtained in [S. Jain and S. Batra, Cyclotomy and arithmetic properties of some families in $\frac{Z[x]}{〈x^{pq}-1〉}$, Asian-Eur. J. Math. 13(4) (2020) 2050077].

Irreducible cyclic codes form an important family of cyclic codes and have applications in space communications. Their weight distributions measure their error performance relative to several channels, and hence have been an interesting object of study for a long time. In this note, we provide a method to determine the weight distributions of q-ary irreducible cyclic codes of length n, where q is a prime power and n is a positive integer coprime to q. This method is more effective for irreducible cyclic codes of some special lengths. We also list some optimal irreducible cyclic codes, which attain the distance bounds given in Grassl's Table [Code Tables: Bounds on the parameters of various types of codes, http://www.codetables.de].