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We use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods., 20 pages, 2 tables. Added information on Cartesian products of graphs. A general reviwe was made, with small additions improving readability. Some typos corrected. arXiv admin note: text overlap with arXiv:1908.08097

We consider generalized Paley graphs $\Gamma(k,q)$, generalized Paley sum graphs $\Gamma^+(k,q)$, and their corresponding complements $\bar \Gamma(k,q)$ and $\bar \Gamma^+(k,q)$, for $k=3,4$. Denote by $\Gamma = \Gamma^*(k,q)$ either $\Gamma(k,q)$ or $\Gamma^+(k,q)$. We compute the spectra of $\Gamma(3,q)$ and $\Gamma(4,q)$ and from them we obtain the spectra of $\Gamma^+(3,q)$ and $\Gamma^+(4,q)$ also. Then we show that, in the non-semiprimitive case, the spectrum of $\Gamma(3,p^{3\ell})$ and $\Gamma(4,p^{4\ell})$ with $p$ prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs $\Gamma(3,p)$ and $\Gamma(4,p)$ for any $\ell \in \mathbb{N}$, respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of $\Gamma^*(k,q)$ such that $\Gamma^*(k,q)$ and $\bar \Gamma^*(k,q)$ are equienergetic for $k=3,4$. In a previous work we have classified all bipartite regular graphs $\Gamma_{bip}$ and all strongly regular graphs $\Gamma_{srg}$ which are complementary equienergetic, i.e.\@ $\{\Gamma_{bip}, \bar{\Gamma}_{bip}\}$ and $\{\Gamma_{srg}, \bar{\Gamma}_{srg}\}$ are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs $\{\Gamma, \bar \Gamma\}$ which are neither bipartite nor strongly regular., Comment: 22 pages

We study the asymptotic behaviour of a class of algebraic geometry codes, which we call block-transitive, that generalizes the classes of transitive and quasi-transitive codes. We prove by using towers of algebraic function fields with either wild or tame ramification, that there are sequences of codes in this family attaining the Tsfasman–Vladut–Zink bound over finite fields of square cardinality. We give the exact length of each code in these sequences as well as explicit lower bounds for their parameters.

We prove that if $H$ is a subgroup of index $n$ of any cyclic group $G$, then $G$ can be isometrically embedded in $(H^n, d_{_{Ham}}^n)$, thus generalizing previous results of Carlet (1998) for $G=\mathbb{Z}_{2^k}$ and Yildiz-\"Ozger (2012) for $G=\mathbb{Z}_{p^k}$ with $p$ prime. Next, for any positive integer $q$ we define the $q$-adic metric $d_q$ in $\mathbb{Z}_{q^n}$ and prove that $(\mathbb{Z}_{q^n}, d_q)$ is isometric to $(\mathbb{Z}_q^n, d_{RT})$ for every $n$, where $d_{RT}$ is the Rosenbloom-Tsfasman metric. More generally, we then demonstrate that any pair of finite groups of the same cardinality are isometric to each other for some metrics that can be explicitly constructed. Finally, we consider a chain $\mathcal{C}$ of subgroups of a given group and define the chain metric $d_{\mathcal{C}}$ and chain isometries between two chains. Let $G, K$ be groups with $|G|=q^n$, $|K|=q$ and let $H, Comment: 25 pages, 2 figures, 3 tables. Some minor typos corrected. Will appear in "Discrete Mathematics"

We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for any abelian group $G$ and any symmetric subset $S$. We then focus on the family of unitary Cayley graphs $G_R=X(R,R^*)$, where $R$ is a finite commutative ring with identity. We show that under mild conditions, $\{G_R, G_R^+\}$ are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that $\{G_R, \bar G_R\}$ are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples $\{G_R, G_R^+, \bar G_R \}$ such that all the graphs are also Ramanujan., Comment: Small typos corrected from v2 (warning: v2 is different from v1, see the comments in v2)

We compute the eta function η ( s ) and its corresponding η-invariant for the Atiyah–Patodi–Singer operator D acting on an orientable compact flat manifold of dimension n = 4 h − 1 , h ≥ 1 , and holonomy group F ≃ Z 2 r , r ∈ N . We show that η ( s ) is a simple entire function times L ( s , χ 4 ) , the L-function associated to the primitive Dirichlet character modulo 4. The η-invariant is 0 or equals ± 2 k for some k ≥ 0 depending on r and n. Furthermore, we construct an infinite family F of orientable Z 2 r -manifolds with F ⊂ SO ( n , Z ) . For the manifolds M ∈ F we have η ( M ) = − 1 2 | T | , where T is the torsion subgroup of H 1 ( M , Z ) , and that η ( M ) determines the whole eta function η ( s , M ) .

We give a reduction formula for the Waring number $g(k,q)$ over a finite field $\mathbb{F}_q$. By exploiting the relation between $g(k,q)$ with the diameter of the generalized Paley graph $\Gamma(k,q)$ and by using the characterization due to Pearce and Praeger (2019) of those $\Gamma(k,q)$ which are Cartesian decomposable, we obtain the reduction formula $$g(\tfrac{p^{ab}-1}{bc},p^{ab}) = b g(\tfrac{p^a-1}{c},p^a)$$ for $p$ prime and $a,b,c$ positive integers under certain arithmetic conditions. Then, we find some arithmetic conditions to apply the formula above, which allow us to obtain many infinite families of explicit values of Waring numbers. Finally, we use the reduction formula together with the characterization of $2$-weight irreducible cyclic codes due to Schmidt and White (2002) to find infinite families of explicit even values of $g(k,q)$., Comment: 26 pages. Some typos solved and minor corrections made. 5 references added

Given a suitable extension $F'/F$ of algebraic function fields over a finite field $\mathbb{F}_q$, we introduce the conorm code $\text{Con}_{F'/F}(\mathcal{C})$ defined over $F'$ which is constructed from an algebraic geometry code $\mathcal{C}$ defined over $F$. We study the parameters of $\text{Con}_{F'/F}(\mathcal{C})$ in terms of the parameters of $\mathcal{C}$, the ramification behavior of the places used to define $\mathcal{C}$ and the genus of $F$. In the case of unramified extensions of function fields we prove that $\text{Con}_{F'/F}(\mathcal{C})^\perp = \text{Con}_{F'/F}({\mathcal{C}}^\perp)$ when the degree of the extension is coprime to the characteristic of $\mathbb{F}_q$. We also study the conorm of cyclic algebraic-geometry codes and we show that some repetition codes, Hermitian codes and all Reed-Solomon codes can be represented as conorm codes., To appear in Advances in Mathematics of Communications

We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over $\mathbb{F}_{q^m}$, and three related codes $\mathcal{C}_{\mathcal{L},0}$, $\mathcal{C}_{\mathcal{L},1}$ and $\mathcal{C}_{\mathcal{L},2}$. We describe the spectra for all these codes when $\mathcal{L}$ is an even rank family, in terms of the distribution of ranks of the forms $Q_R$ in the family $\mathcal{L}$, and we also compute the complete weight enumerator for $\mathcal{C}_\mathcal{L}$. In particular, considering the family $\mathcal{L} = \langle x^{q^\ell} \rangle$, with $\ell$ fixed in $\mathbb{N}$, we give the weight distribution of four parametrized families of cyclic codes $\mathcal{C}_\ell$, $\mathcal{C}_{\ell,0}$, $\mathcal{C}_{\ell,1}$ and $\mathcal{C}_{\ell,2}$ over $\mathbb{F}_q$ with zeros $\{ \alpha^{-(q^\ell+1)} \}$, $\{ 1,\, \alpha^{-(q^\ell+1)} \}$, $\{ \alpha^{-1},\,\alpha^{-(q^\ell+1)} \}$ and $\{ 1,\,\alpha^{-1},\,\alpha^{-(q^\ell+1)}\}$ respectively, where $q = p^s$ with $p$ prime, $\alpha$ is a generator of $\mathbb{F}_{q^m}^*$ and $m/(m,\ell)$ is even. Finally, we give simple necessary and sufficient conditions for Artin-Schreier curves $y^p-y = xR(x) + \beta x$, $p$ prime, associated to polynomials $R \in \mathcal{L}$ to be optimal. We then obtain several maximal and minimal such curves in the case $\mathcal{L} = \langle x^{p^\ell}\rangle$ and $\mathcal{L} = \langle x^{p^\ell}, x^{p^{3\ell}} \rangle$., Comment: 19 pages, 10 tables. We modified the abstract. We included the parameteres of the codes in the theorems (dimensions and minimal distances). We added Lemma 3.2 and Examples 4.3 and 4.5. Also items (ii) in Remark 3.5 and 4.6. Some typos corrected. We added some references also

Let \({\mathcal{D}}\) be the boundary operator defined by Atiyah, Patodi and Singer, acting on smooth even forms of a compact orientable Riemannian manifold M. In continuation of our previous study, we deal with the problem of computing explicitly the η invariant η = η(M) for any orientable compact flat manifold M. After giving an explicit expression for η(s) in the case of cyclic holonomy group, we obtain a combinatorial formula that reduces the computation to the cyclic case. We illustrate the method by determining η(0) for several infinite families, some of them having non-abelian holonomy groups. For cyclic groups of odd prime order p ≥ 7, η(s) can be expressed as a multiple of Lχ(s), an L-function associated to a quadratic character mod p, while η(0) is a (non-zero) integral multiple of the class number h−p of the number field \({\mathbb Q(\sqrt {-p})}\) . In the case of metacyclic groups of odd order pq, with p, q primes, we show that η(0) is a rational multiple of h−p.

We study the spectral theory of the Dirac-type boundary operator $\mathcal{D}$ defined by Atiyah, Patodi, and Singer, acting on smooth even forms of a compact flat Riemannian manifold M. We give an explicit formula for the multiplicities of the eigenvalues of $\mathcal{D}$ in terms of values of characters of exterior representations of SO(n), where n=dim M. As a consequence, we give large families of $\mathcal{D}$ -isospectral flat manifolds that are non-homeomorphic to each other. Furthermore, we derive expressions for the eta series in terms of special values of Hurwitz zeta functions and, as a result, we obtain a simple explicit expression of the eta invariant.

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group \({\mathbb{Z}_p}\), where p is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p = n = 3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.

We obtain a simple formula for the multiplicity of eigenvalues of the Hodge-Laplace operator, Δ f , acting on sections of the full exterior bundle \(\Lambda^{*}(TM) = \bigoplus_{p=0}^n \Lambda^{p}(TM)\) over an arbitrary compact flat Riemannian n-manifold M with holonomy group \({\mathbb{Z}}_2^k\) , 1 ≤ k ≤ n − 1. This formula implies that any two such manifolds having isospectral lattices of translations are isospectral with respect to Δ f . As a consequence, we construct a large family of pairwise Δ f -isospectral and nonhomeomorphic n-manifolds of cardinality greater than \(2^\frac{(n-1)(n-2)}2\) .

Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\Z_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows to compute explicitly the $\eta$-series in terms of values of Riemann-Hurwitz zeta functions, and the $\eta$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\Z_2^k$-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of $\Z_2^k$-manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat $n$-manifolds, pairwise non-homeomorphic to each other., Comment: 39 pages, revised and expanded version, to appear in TAMS

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on p-forms, we determine all p-isospectral and L-isospectral pairs and we show that in this class of manifolds, isospectrality on functions and isospectrality on p-forms for all values of p are equivalent to each other. The list shows for any p, 1 ≤ p ≤ 3, many p-isospectral pairs that are not isospectral on functions and have different lengths of closed geodesics. We also determine all length isospectral pairs (i.e. with the same length multiplicities), showing that there are two weak length isospectral pairs that are not length isospectral, and many pairs, p-isospectral for all p and not length isospectral. Mathematics Subject Classifications (2000): 58J53, 58C22, 20H15.

We give necessary and sufficient conditions for the existence of pin± and spin structures on Riemannian manifolds with holonomy group ℤ2 k . For any n≥4 (resp. n≥6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M 1 , M 2 , non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M 1 admits a pin ± (resp. spin) structure whereas M 2 does not.