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We study the problem of transmission of information over classical and classical-quantum channels in the one-shot regime where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new input probability distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance of group codes in terms of hypothesis testing relative-entropic quantities. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Furthermore, we apply the one-shot result to the asymptotic stationary memoryless setting, and establish a single-letter lower bound on group capacities for both classes of channels. Moreover, we derive a matching upper bound on the asymptotic group capacity., Comment: 41 pages

In this paper, we leverage polar codes and the well-established channel polarization to design capacity-achieving codes with a certain constraint on the weights of all the columns in the generator matrix (GM) while having a low-complexity decoding algorithm. We first show that given a binary-input memoryless symmetric (BMS) channel $W$ and a constant $s \in (0, 1]$, there exists a polarization kernel such that the corresponding polar code is capacity-achieving with the \textit{rate of polarization} $s/2$, and the GM column weights being bounded from above by $N^s$. To improve the sparsity versus error rate trade-off, we devise a column-splitting algorithm and two coding schemes for BEC and then for general BMS channels. The \textit{polar-based} codes generated by the two schemes inherit several fundamental properties of polar codes with the original $2 \times 2$ kernel including the decay in error probability, decoding complexity, and the capacity-achieving property. Furthermore, they demonstrate the additional property that their GM column weights are bounded from above sublinearly in $N$, while the original polar codes have some column weights that are linear in $N$. In particular, for any BEC and $\beta <0.5$, the existence of a sequence of capacity-achieving polar-based codes where all the GM column weights are bounded from above by $N^\lambda$ with $\lambda \approx 0.585$, and with the error probability bounded by $O(2^{-N^{\beta}} )$ under a decoder with complexity $O(N\log N)$, is shown. The existence of similar capacity-achieving polar-based codes with the same decoding complexity is shown for any BMS channel and $\beta <0.5$ with $\lambda \approx 0.631$., Comment: 31 pages, single column. arXiv admin note: substantial text overlap with arXiv:2012.13977

Let $A(n, d)$ denote the maximum size of a binary code of length $n$ and minimum Hamming distance $d$. Studying $A(n, d)$, including efforts to determine it as well to derive bounds on $A(n, d)$ for large $n$'s, is one of the most fundamental subjects in coding theory. In this paper, we explore new lower and upper bounds on $A(n, d)$ in the large-minimum distance regime, in particular, when $d = n/2 - \Omega(\sqrt{n})$. We first provide a new construction of cyclic codes, by carefully selecting specific roots in the binary extension field for the check polynomial, with length $n= 2^m -1$, distance $d \geq n/2 - 2^{c-1}\sqrt{n}$, and size $n^{c+1/2}$, for any $m\geq 4$ and any integer $c$ with $0 \leq c \leq m/2 - 1$. These code parameters are slightly worse than those of the Delsarte--Goethals (DG) codes that provide the previously known best lower bound in the large-minimum distance regime. However, using a similar and extended code construction technique we show a sequence of cyclic codes that improve upon DG codes and provide the best lower bound in a narrower range of the minimum distance $d$, in particular, when $d = n/2 - \Omega(n^{2/3})$. Furthermore, by leveraging a Fourier-analytic view of Delsarte's linear program, upper bounds on $A(n, n/2 - \rho\sqrt{n})$ with $\rho\in (0.5, 9.5)$ are obtained that scale polynomially in $n$. To the best of authors' knowledge, the upper bound due to Barg and Nogin \cite{barg2006spectral} is the only previously known upper bound that scale polynomially in $n$ in this regime. We numerically demonstrate that our upper bound improves upon the Barg-Nogin upper bound in the specified high-minimum distance regime., Comment: 23 pages

In this paper, we study codes with sparse generator matrices. More specifically, low-density generator matrix (LDGM) codes with a certain constraint on the weight of the columns in the generator matrix are considered. In this paper, it is first shown that when a BMS channel W and a constant s>0 are given, there exists a polarization kernel such that the corresponding polar code is capacity-achieving and the column weights of the generator matrix (GM) are bounded from above by $N^s$. Then, a general construction based on a concatenation of polar codes and a rate-$1$ code, and a new column-splitting algorithm that guarantees a much sparser GM, is given. More specifically, for any BMS channel and any $\epsilon > 2\epsilon^*$, where $\epsilon^* \approx 0.085$, an existence of a sequence of capacity-achieving codes with all the GM column weights upper bounded by $(\log N)^{1+\epsilon}$ is shown. Furthermore, two coding schemes for BEC and BMS channels, based on a second column-splitting algorithm, are devised with low-complexity decoding that uses successive-cancellation. The second splitting algorithm allows for the use of a low-complexity decoder by preserving the reliability of the bit-channels observed by the source bits, and by increasing the code block length. The concatenation-based construction can also be applied to the random linear code ensemble to yield capacity-achieving codes with all the GM column weights being $O(\log N)$ and with (large-degree) polynomial decoding complexity., Comment: Extended version, now includes moderate-block length comparison with the RLE. arXiv admin note: text overlap with arXiv:2001.11986

In this paper we study codes with sparse generator matrices. More specifically, codes with a certain constraint on the weight of all the columns in the generator matrix are considered. The end result is the following. For any binary-input memoryless symmetric (BMS) channel and any epsilon > 2 epsilon*, where epsilon^* = \frac{1}{6}-\frac{5}{3}\log{\frac{4}{3}} \approx 0.085, we show an explicit sequence of capacity-achieving codes with all the column wights of the generator matrix upper bounded by (\log N)^{1+epsilon}, where N is the code block length. The constructions are based on polar codes. Applications to crowdsourcing are also shown., Comment: 12 pages, 2 tables

This paper models the crowdsourced labeling/classification problem as a sparsely encoded source coding problem, where each query answer, regarded as a code bit, is the XOR of a small number of labels, as source information bits. In this paper we leverage the connections between this problem and well-studied codes with sparse representations for the channel coding problem to provide querying schemes with almost optimal number of queries, each of which involving only a constant number of labels. We also extend this scenario to the case where some workers can be unresponsive. For this case, we propose querying schemes where each query involves only log n items, where n is the total number of items to be labeled. Furthermore, we consider classification of two correlated labeling systems and provide two-stage querying schemes with almost optimal number of queries each involving a constant number of labels., Comment: 6 pages

While research concerning feedback and cooperative learning is prevalent in educational literature, there has been little systematic research to support whether various types of feedback in cooperative learning situations can affect student performance. This study examines possible interactions among different types of computer-delivered feedback on cognitive learning outcomes and attitudes during computer-based cooperative condition. The achievement of high and low-prior knowledge students was compared among different feedback treatments; the subjects were 120 university students enrolled in six sections of beginning weight training at the University of Minnesota. The computer-based instructional module was designed to help the beginners build their basic cognitive foundation in weight training; a retention test and an attitude questionnaire were administered to the students. Results indicated significant interaction between types of feedback and prior knowledge level; results on attitude measure revealed that low-prior knowledge students had more positive attitudes than high-prior knowledge students. (Contains 13 references.) (AEF)

Computer-based instruction opens new avenues for increasing the variety of possible feedback strategies that a teacher may employ to optimize learner performance. Feedback effectiveness is influenced by the nature of learning task and student ability. This study investigates the effects of digitized feedback and ability on the achievement of college students during computer-based instruction; the achievement of high and low-prior knowledge students was compared among different feedback treatments. A sample of 68 university students from four sections of beginning tennis classes at Chung Cheng University (Taiwan) were categorized as having high or low prior knowledge and randomly assigned to a computer to complete one of three treatments (audio only, voice with text, and voice with text and animation). The computer-based instructional unit was an interactive video lesson on cognitive areas of tennis skill performance. Results indicate that the anticipated interaction between ability and feedback was not confirmed, and that elaborate feedback was most beneficial for cognitive ares of psychomotor skill learning; no matter which ability levels or treatment conditions the subjects were categorized in, they rated the instructional program more positively. (Contains 14 references.) (AEF)