Cyclic simplex codes are usually deemed as impractical because of their extremely low rate and large codeword length for practical values of the code dimension. To address these limitations, researchers have resorted to punctured simplex codes, focusing on primitive polynomials and employing statistical analyses to investigate their properties. This paper delves deeper into the properties of punctured binary simplex codes, also focusing on the recently introduced family of Primitive Rate-Compatible Low-Density Parity-Check (PRC-LDPC) codes. We study the average behavior of punctured simplex codes in terms of minimum distance properties. Furthermore, our results highlight the potential of high-rate PRC-LDPC codes to reach or even surpass the performance of state-of-the-art code families. We show how to design good codes by applying puncturing and shortening operations to cyclic simplex codes, also considering complexity-constrained scenarios.

In this paper we study a special family of Low-Density Parity-Check (LDPC) codes, called Primitive Rate-Compatible (PRC) LDPC codes, obtained by applying puncturing to some properly chosen simplex codes. We investigate their minimum distance properties and give insights on the design of PRC-LDPC codes for use in scenarios where constraints on hardware and software resources require maintaining a low level of complexity. We numerically assess the error rate performance of these codes in such a setting, showing that they also demonstrate good performance under belief propagation decoding, in addition to low decoding complexity.

We introduce and study a family of rate-compatible Low-Density Parity-Check (LDPC) codes. The design of these codes starts from simplex codes, defined by parity-check matrices having a simple form stemming from the coefficients of a primitive polynomial. For this reason, we call the new codes Primitive Rate-Compatible LDPC (PRC-LDPC) codes. By applying puncturing to these codes, we obtain a bit-level granularity of the code rate. We show that, in order to achieve good LDPC codes, the underlying polynomials, besides being primitive, must meet some more stringent conditions with respect to those of classical punctured simplex codes. We leverage non-modular Golomb rulers to take these new requirements into account. We characterize the minimum distance properties of PRC-LDPC codes, and study and discuss their encoding and decoding complexity. Finally, we assess the error rate performance of high rate PRC-LDPC codes under iterative decoding.

In this paper we show that, when a binary primitive polynomial can be associated to a sparse Golomb ruler, the simplex code obtained by taking it as the code parity-check polynomial exhibits good distance properties and performance. We define some conditions under which the obtained codes are also Low-Density Parity-Check (LDPC) codes, and can hence be decoded through efficient iterative algorithms. We perform code puncturing, leading to a family of rate-adaptive codes, and we predict some of their structural properties in terms of minimum distance and weight distribution. We show that, in addition to having some useful properties, these codes achieve good performance in terms of error rate under LDPC decoding.

In this paper we study recent reaction attacks against QC-LDPC and QC-MDPC code-based cryptosystems, which allow an opponent to recover the private parity-check matrix through its distance spectrum by observing a sufficiently high number of decryption failures. We consider a special class of codes, known as monomial codes, to form private keys with the desirable property of having a unique and complete distance spectrum. We verify that for these codes the problem of recovering the secret key from the distance spectrum is equivalent to that of finding cliques in a graph, and use this equivalence to prove that current reaction attacks are not applicable when codes of this type are used in the McEliece cryptosystem.

In this paper, time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs) are considered, and the connections existing between their low-weight codewords and cycles in their Tanner graphs are studied. Using the polynomial representation of these codes, we show that parity-check matrices having columns with weight ≥2 can be analyzed considering a certain number of parity-check sub-matrices having regular columns with weight 2. These sub-matrices are associated to cycles in the code Tanner graph and define as many codes we denote as component codes. Based on this observation, we find that codewords of the main code can be expressed as combinations of codewords of the component codes. The design of codes free of codewords up to a certain weight is also addressed. We show that low-weight codewords in the main code can be avoided by removing some types of cycles in its Tanner graph. Our design approach is applied to some well-known ensembles of SC-LDPC-CCs to prove its effectiveness.

In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.

Product LDPC codes take advantage of LDPC decoding algorithms and the high minimum distance of product codes. We propose to add suitable interleavers to improve the waterfall performance of LDPC decoding. Interleaving also reduces the number of low weight codewords, that gives a further advantage in the error floor region.

In this paper, the basic theory of interleavers is revisited in a semi-tutorial manner, and extended to encompass noncausal interleavers. The parameters that characterize the interleaver behavior (like delay, latency, and period) are clearly defined. The input-output interleaver code is introduced and its complexity studied. Connections among various interleaver parameters are explored. The classes of convolutional and block interleavers are considered, and their practical implementation discussed. The trellis complexity of turbo codes is tied to the complexity of the constituent interleaver. A procedure of complexity reduction by coordinate permutation is also presented, together with some examples of its application.

An investigation is presented of the maximum transmission efficiency that can be reached over an ideal photon counting channel, having fixed the bandwidth expansion factor. First, the ideal situation, represented by C.E. Shannon's (1959) theorem for discrete channels, is analyzed. A low average number of photons per pulse is demonstrated to be preferable. A binary transmission, with different a priori probabilities of the two transmitted symbols, exhibits a higher efficiency than that of an orthogonal PPM (pulse-position modulated) transmission, whose M-ary symbols are equiprobable, for an equal bandwidth expansion. Then practical transmissions are considered. The PPM technique can be very efficiently coded, and in some situations, is characterized by a bit error probability lower than that of the uncoded binary technique. However, uncoded binary transmission remains extremely attractive for the achievement of ultrahigh transmission efficiencies.>