The vastness of the IPv6 address space has led to the common practice of allocating prefixes to end users rather than individual addresses. Users can assign these prefixes as a single subnet or divide them into multiple subnets for different purposes. Allocation strategies vary significantly in terms of prefix granularity, and identifying the actual granularity of subnet assignments is crucial for improving measurement efficiency, accuracy, and for better IPv6 network management. However, no existing method can discover IPv6 subnets at an Internet-Wide scale.To this end, we propose SubRecon, an Internet-Wide IPv6 subnet discovery system. SubRecon consists of two key phases: subnet delimitation and target expansion. In the subnet delimitation phase, we perform a systematic scan across the entire IPv6 address space without relying on any existing seed dataset. This phase adopts a top-down approach, probing prefixes from the shortest to the longest in a hierarchical manner. At each level, we recursively refine prefixes and discard sub-prefixes that do not meet the convergence condition. This pruning strategy eliminates redundant probes in unallocated regions, significantly reducing the search space and improving probing efficiency. To further improve coverage, the target expansion phase leverages the active address dataset as an auxiliary input. It identifies active addresses not covered by previously discovered subnets, expands them into new candidate target prefixes, and performs another round of subnet delimitation. This helps enhance the completeness and coverage of the final discovered subnet set. Experimental results show that SubRecon discovers 8,381,974 IPv6 subnets across 14,147 autonomous systems, and the resulting subnet list can serve as high-quality input for topology discovery. Additionally, during the subnet discovery process, SubRecon identifies a large number of last-hop router interfaces, discovering approximately 10 million more than the current state-of-the-art methods.

It is proved in a reference (Fan, Lin, IEEE TIT, vol.67, pp.5016-5025) that the self-dual (LCD respectively) dihedral codes over a finite fieldFwith$|F|=q$are asymptotically good ifqis even (odd respectively). In this paper, we investigate the algebraic structures and the asymptotic properties of consta-dihedral codes overF, and show that: ifqis even or$4\,|\,(q-1)$, then the self-dual consta-dihedral codes are asymptotically good; otherwise, the LCD consta-dihedral codes are asymptotically good. And, with the help of a technique developed in this paper, some errors in the reference mentioned above are corrected.

Let F be a field with cardinality$p^{\ell } $and$0\neq \lambda \in F$, and$0\le h\lt \ell $. Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois$p^{h}$-inner product (DCC, vol.84, pp.473–492). In this paper, we characterize the structure of 2-quasi$\lambda $-constacyclic codes over F; and exhibit necessary and sufficient conditions for 2-quasi$\lambda $-constacyclic codes being Galois$p^{h}$-self-dual. With the help of a technique developed in this paper, we prove that, when$\ell $is even, the Hermitian self-dual 2-quasi$\lambda $-constacyclic codes are asymptotically good if and only if$\lambda ^{1+p^{\ell /2}}\!=1$. And, when$p^{\ell } \,{\cancel {\equiv }}\,3~({\mathrm { mod}}~4)$, the Euclidean self-dual 2-quasi$\lambda $-constacyclic codes are asymptotically good if and only if$\lambda ^{2}=1$.

Many kinds of codes which possess two cycle structures over two special finite commutative chain rings, such as${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$-additive cyclic codes and quasi-cyclic codes of fractional index etc., were proved asymptotically good. In this paper we extend the study in two directions: we consider any two finite commutative chain rings with a surjective homomorphism from one to the other, and consider double constacyclic structures. We construct an extensive kind of double constacyclic codes over two finite commutative chain rings. And, developing a probabilistic method suitable for quasi-cyclic codes over fields, we prove that the double constacyclic codes over two finite commutative chain rings are asymptotically good.

A class of self-dual quasi-abelian codes of index 2 over any finite field$F$is introduced. By counting the number of such codes and the number of the codes in this class whose relative minimum weights are small, such codes are proved to be asymptotically good provided −1 is a square in$F$. Moreover, a class of self-orthogonal quasi-abelian codes of index 2 is defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that these self-orthogonal quasi-abelian codes of index 2 are asymptotically good.

Bazzi and Mitter showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with strong duality property are asymptotically good. If the characteristic of the field is even, self-dual dihedral group codes are asymptotically good. If the characteristic of the field is odd, maximal self-orthogonal dihedral group codes and LCD dihedral group codes are asymptotically good.

We introduce quasi-cyclic codes of index 1(1/3), and construct a class of such codes generated by pairs of polynomials. By investigating the pair of circulant matrices associated with the generator pair of polynomials, we obtain the generator matrix of any code of the class. Using a probabilistic method, we prove that, for any positive real number δ such that the asymptotic GV-bound at 2δ is greater than 1/2, the probability that the relative minimal distance of the code in the class is greater than δ is almost 1; and the probability that the rate of the code equals to 1/4 is also almost 1. An obvious consequence is that the quasicyclic codes of index 1(1/3) are asymptotically good.

For any given positive integer m, a necessary and sufficient condition for the existence of Type-I m-adic constacyclic codes is given. Furthermore, for any given integer s, a necessary and sufficient condition for s to be a multiplier of a Type-I polyadic constacyclic code is given. As an application, some optimal codes from Type-I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant maximum distance separable codes, are constructed.

For a q-ary random quasi-Abelian code with fixed coindex and constant rate r, it is shown that the Gilbert-Varshamov (GV)-bound is a threshold point: if r is less than the GV-bound at δ ∈ (0, 1 - q-1), then the probability of the relative distance of the random code being greater than δ approaches 1 as the index goes to infinity; whereas, if r is bigger than the GV-bound at δ, then the probability approaches 0. As a corollary, there exist numerous asymptotically good quasi-Abelian codes attaining the GV-bound.

We begin by reviewing the major results dealing with the structure of the cl-degrees, and then focus on the method of the Yu–Ding Theorem, which is an important result in this area. By strengthening the Yu–Ding procedure, we construct a cl-cuppable cl-degree of c.e. sets.

Regarding any finite field as a residue field of the algebraic integer ring of a cyclotomic field, we select a system of representatives in the ring with minimal Manhattan metric, and introduce a Mannheim weight on the finite field. The linear codes over the finite field with the Mannheim weight are discussed. A geometric method to compute the representatives in Gaussian integers is provided.