In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain $\mathbb{F}_q[t]/(t^i)$, to composite same alphabets. We define infinite ring $R_\infty$ as ideals in group $R_\infty G.$ show that dual of a $G$-code is again setting. known results projections lifts $G$-codes. Additionally, some $\gamma$-adic study these codes principal ideal rings.

In this paper we give the structure of constacyclic codes over formal power series and chain rings. We also present necessary and sufficient conditions on the existence of MDS codes over principal ideal rings. These results allow for the construction of infinite families of MDS self-dual codes over finite chain rings, formal power series and principal ideal rings.

A (variable-length) code is a free subset of a free monoid; more formally, it is the basis of some free submonoid. The theory of these codes (not to be confused with error-detecting codes) was first developed by M.P. Schiitzenberger. A major problem in this theory is to factorize codes, that is, considering the characteristic formal power series of a code (in the noncommutat ive formal power se...

Formal power series in N non-commuting indeterminates can be considered as a counterpart of functions of one variable holomorphic at 0, and some of their properties are described in terms of coefficients. However, really fruitful analysis begins when one considers for them evaluations on N-tuples of n× n matrices (with n = 1, 2, . . .) or operators on an infinite-dimensional separable Hilbert s...

Abstract. Perturbative quantum field theory associates formal power series invariants to knotted objects, that is to knots or homology 3-spheres. These formal power series are known to be Gevrey, and are expected to be factorially divergent, and somehow linked to quantum invariants of knotted objects at complex roots of unity. The latter are the well-known Witten-Reshetikhin-Turaev invariants o...

This paper concerns Laurent series solutions of algebraic ordinary differential equations (AODEs). We first present several approaches to compute formal power series solutions of a given AODE. Then we determine a bound for orders of its Laurent series solutions. Using the order bound, one can transform a given AODE into a new one whose Laurent series solutions are only formal power series. The ...

This article presents a flatness–based approach to the trajectory planning and feedforward control problem for the inviscid Burger equation with and without an additional quadratic nonlinearity. It uses the property of formal power series parameterizability of the underlying partial differential equation and uniform Euler–summability of the resulting power series to derive a parameterization of...