Non-unital Rings Research Articles

Cyclic Codes over a Non-Commutative Non-Unital Ring

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E=⟨a,b∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over E. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length 2n over F2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.

Cyclic Codes over a Non-Local Non-Unital Ring

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Self-orthogonal codes over a non-unital ring from two class association schemes

There is a non-unital ring [Formula: see text] of order 4 defined by generators and relations as [Formula: see text] In this paper, we present special constructions of linear codes over [Formula: see text] from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tournaments (DRTs). We investigate the conditions under which these codes are self-orthogonal, quasi self-dual, or Type IV. As a byproduct of this study, some diophantine relations on the weight of sum of vectors with coefficients in [Formula: see text] are derived. Some examples of codes with minimum distance better than that of Type IV codes over unital rings of the same order in modest lengths are given.

Mass Formula for Self-Orthogonal and Self-Dual Codes over Non-Unital Rings of Order Four

We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring I=a,b|2a=2b=0,a2=b,ab=0 and the noncommutative ring E=a,b|2a=2b=0,a2=a,b2=b,ab=a,ba=b. We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.

On the classification of codes over non-unital ring of order 4

In the last 60 years coding theory has been studied a lot over finite fields [Formula: see text] or commutative rings [Formula: see text] with unity. Although in [Formula: see text], a study on the classification of the rings (not necessarily commutative or ring with unity) of order [Formula: see text] had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring [Formula: see text] by presenting the classification of optimal and nice codes of length [Formula: see text] over [Formula: see text], along with respective weight enumerators and complete weight enumerators.

Minimal and optimal binary codes obtained using $$C_D$$-construction over the non-unital ring I

Minimal and optimal binary codes obtained using $$C_D$$-construction over the non-unital ring I

Correction: Self-orthogonal codes over a non-unital ring and combinatorial matrices

Correction: Self-orthogonal codes over a non-unital ring and combinatorial matrices

Quasi self-dual codes over non-unital rings from three-class association schemes

<abstract><p>Let $ E $ and $ I $ denote the two non-unital rings of order 4 in the notation of (Fine, 93) defined by generators and relations as $ E = \langle a, b \mid 2a = 2b = 0, a^2 = a, b^2 = b, ab = a, ba = b\rangle $ and $ I = \langle a, b \mid 2a = 2b = 0, a^2 = b, ab = 0\rangle $. Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the rings $ E $ and $ I $ for lengths up to 12 and 6, respectively. The codes had minimum distance at most 2 in the case of $ I $, and 4 in the case of $ E $. In this paper, we present two methods for constructing linear codes over these two rings using the adjacency matrices of three-class association schemes. We show that under certain conditions the constructions yield QSD or Type Ⅳ codes. Many codes with minimum distance exceeding 4 are presented. The form of the generator matrices of the codes with these constructions prompted some new results on free codes over $ E $ and $ I $.</p></abstract>

Duality of Codes over Non-unital Rings of Order Four

In this paper, we present a basic theory of the duality of linear codes over three of the non-unital rings of order four; namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I, E</i> , and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> as denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> . The notion of self-dual codes with respect to this duality coincides with that of quasi self-dual codes over <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> as introduced in (Alahmadi <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> , 2022). We characterize self-dual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> . We study the connection between the dual of any linear code over these rings and the dual of its associated binary codes. A MacWilliams formula is established for linear codes over <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> .

The mass formula for self-orthogonal and self-dual codes over a non-unitary commutative ring

&lt;abstract&gt;&lt;p&gt;In this paper, we establish a mass formula for self-orthogonal codes, quasi self-dual codes, and self-dual codes over commutative non-unital rings $ {{\mathit {I}_p}} = \left &amp;lt; a, b | pa = pb = 0, a^2 = b, ab = 0 \right &amp;gt; $, where $ p $ is an odd prime. We also give a classification of the three said classes of codes over $ {{\mathit {I}_p}} $ where $ p = 3, 5, $ and $ 7 $, with lengths up to $ 3 $.&lt;/p&gt;&lt;/abstract&gt;

The build-up construction over a commutative non-unital ring

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as $$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^2=b,\, \,ab=0 \rangle . \end{aligned}$$We study a recursive construction of self-orthogonal codes over I. We classify self orthogonal codes of length n and size \(2^n\) (called here quasi self-dual codes or QSD) up to the length \(n=6.\) In particular, we classify Type IV codes (QSD codes with even weights) and quasi Type IV codes (QSD codes with even torsion code) up to \(n=6.\)

Construction of quasi self-dual codes over a commutative non-unital ring of order 4

Construction of quasi self-dual codes over a commutative non-unital ring of order 4

Cancellation properties of graded and nonunital rings. Graded clean and graded exchange Leavitt path algebras

Various authors have been generalizing some unital ring properties to nonunital rings. We consider properties related to cancellation of modules (being unit-regular, having stable range one, being directly finite, exchange, or clean) and their “local” versions. We explore their relationships and extend the defined concepts to graded rings. With graded clean and graded exchange rings suitably defined, we study how these properties behave under the formation of graded matrix rings. We exhibit properties of a graph E which are equivalent to the unital Leavitt path algebra [Formula: see text] being graded clean. We also exhibit some graph properties which are necessary and some which are sufficient for [Formula: see text] to be graded exchange.

LCD and ACD codes over a noncommutative non-unital ring with four elements

We study LCD (linear complementary dual) and ACD (additive complementary dual) codes over a noncommutative non-unital ring E with four elements. This is the first attempt to construct LCD codes over a noncommutative non-unital ring. We show that free LCD codes over E are directly related to binary LCD codes. We introduce ACD codes over E. They include free LCD codes over E as a special case. These facts imply that LCD and ACD codes over E are worth studying. In particular, we characterize a free LCD E-code C in terms of a binary generator matrix G. We also define an ACD code over E, called a left-ACD code. We give several conditions for the existence of left-ACD codes.

Self-orthogonal codes over a non-unital ring and combinatorial matrices

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \(\mathbb {F}_4.\) The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.

The build-up construction of quasi self-dual codes over a non-unital ring

There is a local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by generators and relations as [Formula: see text] We study a recursive construction of self-orthogonal codes over [Formula: see text] We classify, up to permutation equivalence, self-orthogonal codes of length [Formula: see text] and size [Formula: see text] (called here quasi self-dual codes or QSD) up to the length [Formula: see text]. In particular, we classify Type IV codes (QSD codes with even weights) up to [Formula: see text].

A note on the right-left symmetry of aR ⊕ bR = (a + b)R in rings

We give an example to show that, for nonunital rings [Formula: see text], the direct sum [Formula: see text] with [Formula: see text] regular has no in general right-left symmetry. It is then proved that the right-left symmetry actually holds in a left and right faithful ring.

Quasi self-dual codes over non-unital rings of order six

There exist two semi-local rings of order 6 without identity for the multiplication. We classify up to coordinate permutation self-orthogonal codes of length n and size 6n/2 over these rings (called here quasi self-dual codes or QSD) till the length n = 8. To any such code is attached canonically a ℤ6-code, which, when self-dual, produces an unimodular lattice by Construction A.

Multiplier Hopf algebras: Globalization for partial actions

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.

A “quite superfluous” characterization of the Jacobson radical of an associative ring

It is well-known that the Jacobson radical of a unital ring R is the largest superfluous right ideal of R. While a simple example shows this to be false for non-unital rings, it is shown that the result remains valid in the absence of a unit provided the notion of “superfluous” is taken relative to all regular right ideals. The purpose of this note is to provide a proof for this little-known fact.