Detecting the integer parameter vector in a box-constrained linear model with additive Gaussian noise arises from many applications. The maximum likelihood (ML) detector detects the integer parameter vector by solving a Box-constrained Integer Least Squares (BILS) problem and achieves the highest success probability (i.e., the probability that the detected integer vector equals the original integer vector). However, due to the high complexity of the ML detector, the Babai detector is frequently used to approximate solutions to the BILS problem, especially in time-constrained applications. Considering the success probability achieved by existing Babai detector frameworks is not always satisfying, this paper focuses on proposing a fast block-based Babai detector framework within polynomial complexity to narrow the gap to the ML detector. Unlike element-wise Babai detectors, the proposed method partitions the model matrix into blocks and applies the appropriately sized ML detector to each block. Theoretical analysis shows that the success probability of the block-based Babai detector increases with the block size and therefore is better than the Babai detector. To further improve the success probability, an extended block-based Babai detector is proposed with a reasonable increase in complexity. Simulation results illustrate the theoretical findings and indicate that the proposed detectors achieve a better trade-off between the success probability and complexity compared with conventional Zero-Forcing (ZF), Successive Interference Cancellation (SIC)/Babai detectors, as well as the generalized Babai detector proposed by Chang et al.. For example, in a 16 × 16 MIMO system with 16-QAM modulation, the (extedned) block-based Babai detector can improve the success probability by 5.3% (8.7%) at SNR = 15dB and achieve a gain of approximately 1.5 dB (2 dB) at a BER of 10−3, compared with Chang et al.’s generalized Babai detector, while requiring only 45.1% (57.9%) of the computational complexity on average over the tested SNR range.

The Assmus-Mattson Theorem is a famous and effective way to constructt-designs from supports of codewords of a linear code. By extending this idea, we study thet-designs constructed from supports of subcodes of linear codes. In this paper, we introduce the notion ofr-generalized maximum distance separable codes (r-GMDS codes) and determine thet-designs constructed from supports of subcodes of ar-GMDS code. Also, we show thatt-designs can be constructed from supports of subcodes of a linear code if the automorphism group of this linear code ist-transitive. In particular, we prove thatt-designs can be constructed from supports of subcodes of Reed-Muller codes. Then we show that the number of blocks in thet-design constructed from supports of subcodes of a linear code is related to the subcode support weight distributions of this linear code. Next, we provide some new formulas for the subcode support weight distributions ofr-GMDS codes. After we present a connection betweenl-MDS codes andr-GMDS codes, some new formulas are obtained for the subcode support weight distributions ofl-MDS codes. In particular, we completely determine the subcode support weight distributions of maximum distance separable (MDS) codes and near maximum distance separable (NMDS) codes. As an example, the subcode support weight distributions of the extended ternary Golay code are completely determined.

The favorable algebraic properties and error-correcting performance of BCH codes have motivated extensive research and diverse applications, whereas the study of extended BCH codes remains relatively less explored. In this paper, we focus on the extended BCH codeC(q,q+1,δ,h)over the finite field Fq. Specifically, we investigate the parameters of three families of extended BCH codes and the weight distributions of their duals by solving certain quadratic and quartic equations:C(q,q+1,3,1)is shown to be NMDS when gcd(q+1,3) = 1;C(q,q+1,δ,q)with 3 ≤ δ ≤ ⌈(q+ 5)/2⌉ is proved to be AMDS; andC(q,q+1,3,q/2)is MDS for evenq.

Abstract-In this paper, we study a class of twisted generalized Reed-Solomon (TGRS) codes with general$\ell$twists. A sufficient and necessary condition for the TGRS codes to be MDS or$\ell$-MDS$(\ell < \min \{k, n-k\})$is determined. A sufficient and necessary condition that such a TGRS code is self-dual for$\ell \leq\left\lfloor\frac{k-1}{3}\right\rfloor$is also presented. Finally, we give an explicit construction of self-dual TGRS codes.

List-decoding and average-radius list-decoding are important generalizations of unique decoding that received considerable attention over the years. However, the optimal trade-off among list-decoding radius, list size, and the code rate are not fully understood in both problems. In this paper, firstly, we prove a new Singleton-type bound for list-decodability, which improves the results of [3]. Next, we prove a Singleton-type bound for average-radius list-decodable codes, which is to the best of our knowledge, the first such bound for average-radius list-decodable codes.

We give a method to construct deep holes for elliptic curve codes. For long elliptic curve codes, we conjecture that our construction is complete in the sense that it gives all deep holes. Some evidence and heuristics on the completeness are provided by means of connections with problems and results in finite geometry.

Multi-spectral pedestrian detection has gained extensive attention over the past decade. To alleviate the problem of modality imbalance in the multi-spectral tasks, a novel cross-guided feature fusion network based on the auto-encoder framework is proposed using RGB-thermal image pairs as inputs. To obtain the complementary features, a cross-guided loss is designed, so that the output images are balanced with both modalities in an unsupervised manner. An intra-modality reweighting module is implemented to filter the redundant features before the fusion. Finally, YOLOv3 is chosen as the detector fed by the fused features. The proposed method is verified using the public KAIST and VOT-RGBT datasets. Experimental results demonstrate that the proposed method can outperform the state-of-the-art methods, the miss rate of pedestrian detection reaches 48.57% and 4.52% using KAIST and VOT-RGBT datasets, respectively.

Locally repairable codes (LRCs) are important for distributed storage systems due to their efficient repairing ability of the failed storage nodes. A$q$-ary optimal$(n,k,r)$-LRC is an$[n,k,d]$linear code over$\mathbb {F}_{q}$such that every code symbol has locality$r$, and the minimum distance attains the well-known Singleton-like bound. In this paper, we study the maximal code length, code constructions and weight distributions of$q$-ary optimal LRCs with locality 2 and distance 5, which are of both practical and theoretical interest. Firstly, it is proved that when the code dimension is even or odd, corresponding maximal code lengths of such$q$-ary optimal LRCs are$3 \cdot \lfloor \frac {q+1}{3} \rfloor $and$3 \cdot \left \lfloor{ \frac {q-1}{3} }\right \rfloor +5$, respectively. Up to the equivalence of linear codes, we propose constructions of all the possible$q$-ary optimal LRCs with locality 2, distance 5 and maximal code length. Then, by characterizing the weight type hierarchy of codewords, we show that the weight distribution of any$q$-ary optimal LRC with locality 2, distance 5 and even code dimension can be uniquely determined and explicit expression of the weight distribution is given. Moreover, it is shown that all$q$-ary optimal LRCs with locality 2, distance 5 and even code dimension are maximally recoverable.

A Goppa code over \Bbb Fqmis a well-known subclass of algebraic error-correcting code. If m=1, then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters.

The weight distribution of a q-ary [n, k, d] linear code is an important research subject in coding theory. In a linear code, a code symbol is said to have locality r if it can be recovered by accessing at most r other code symbols. A q-ary locally repairable code (LRC) is an [n, k, d] linear code over Fq such that every code symbol has locality r, and is said to be optimal if the minimum distance attains the well-known Singleton-like bound. In this paper, we focus on the weight distributions of q-ary optimal LRCs with locality 2, minimum distance 5 and even dimension k. By analyzing the parity-check matrices involving locality, it is shown that the weight distributions of all q-ary optimal LRCs with locality 2, distance 5, even dimension k and code length n can be uniquely determined and explicit expressions of the weight distributions are given.

Projective Reed-Solomon (PRS) codes are Reed-Solomon codes of the maximum possible length q+1. The classification of deep holes-received words with maximum possible error distance- for PRS codes is an important and difficult problem. In this paper, we use algebraic methods to explicitly construct three classes of deep holes for PRS codes. We show that these three classes completely classify all deep holes of PRS codes with redundancy four. Previously, the deep hole classification was only known for PRS codes with redundancy at most three.

We determine the 2-adic complexity of the Ding-Helleseth-Martinsen (DHM) binary sequences by using cyclotomic numbers of order four, “Gauss periods” and “quadratic Gauss sums” on finite field Fq and valued in Z2N-1, where q ≡ 5 (mod 8) is a prime number and N = 2q is the period of the DHM sequences.

In this paper, we obtain new results on the covering radius and deep holes for projective Reed-Solomon (PRS) codes.

Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [1] showed that it is an NP-hard problem for general linear codes. In practice, one often uses codes with additional mathematical structure, such as cyclic codes and algebraic geometry (AG) codes, etc. In this paper, we study the minimum distance of a family of AG codes. For AG codes of genus 0 (generalized Reed-Solomon codes), the minimum distance has a simple explicit formula. An interesting result of Cheng [2] says that the minimum distance problem is already NP-hard (under RP-reduction) for general elliptic curve codes (ECAG codes, or AG codes of genus 1). In this paper, we show that the minimum distance of ECAG codes also has a simple explicit formula if the evaluation set is suitably large (at least 2=3 of the group order). Our method is purely combinatorial and based on a new sieving technique from Li-Wan [3].

The network coding system COPE [1] shows us the improvements obtained by COPE-type network coding in wireless networks. In this paper, by introducing 802.11 interference model, we analyze the maximal multi-commodity flow problem in wireless network coding and put forward a new scheduling mechanism 802.11st which has higher throughput for multihop wireless networks with network coding. We use linear programming to compute the maximum multi-commodity flow of multiple unicast flows in which the constraints are rebuilt according to the new scheduling mechanism.

Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n, k] linear code over Fqwith parity-check matrix H, where the rows of H may be dependent. Let [n] = {1, 2,...,n} denote the set of column indices of H. A stopping set S of C with parity-check matrix H is a subset of [n] such that the restriction of H to S does not contain a row of weight 1. The stopping set distribution {Ti(H)}i=0nenumerates the number of stopping sets with size i of C with parity-check matrix H. Denote H*, the parity-check matrix, consisting of all the nonzero codewords in the dual code C⊥. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix H*. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed-Solomon codes, it is easy to determine all the stopping sets. Then, we consider the AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then, the stopping sets, the stopping set distribution, and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting, and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of the AG codes from elliptic curves.