We investigate fat projective linear codes over${\mathbb Z}_{p^{m}}$,$m\geqslant 2$, with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator matrix of the code determine a cap in the projective Hjelmslev geometry$\mathop {\mathrm {PHG}}\nolimits (k-1, {\mathbb Z}_{p^{m}})$. This generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982). The proof relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry.

It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 16, 30, 31, 32, 45-51, and ≥ 60.

Linear encoders with good joint spectra are suitable candidates for optimal lossless joint source-channel coding (JSCC), where the joint spectrum is a variant of the input-output complete weight distribution and is considered good if it is close to the average joint spectrum of all linear encoders (of the same coding rate). In spite of their existence, little is known on how to construct such encoders in practice. This paper is devoted to their construction. In particular, two families of linear encoders are presented and proved to have good joint spectra. The first family is derived from Gabidulin codes, a class of maximum-rank-distance codes. The second family is constructed using a serial concatenation of an encoder of a low-density parity-check code (as outer encoder) with a low-density generator matrix encoder (as inner encoder). In addition, criteria for good linear encoders are defined for three coding applications: 1) lossless source coding; 2) channel coding; and 3) lossless JSCC. In the framework of the code-spectrum approach, these three scenarios correspond to the problems of constructing linear encoders with good kernel spectra, good image spectra, and good joint spectra, respectively. Good joint spectra imply both good kernel spectra and good image spectra, and for every linear encoder having a good kernel (respectively, image) spectrum, it is proved that there exists a linear encoder not only with the same kernel (respectively, image) but also with a good joint spectrum. Thus, a good joint spectrum is the most important feature of a linear encoder.

Recent research indicates that packet transmission employing random linear network coding can be regarded as transmitting subspaces over a linear operator channel (LOC). In this paper we propose the framework of linear operator broadcast channels (LOBCs) to model packet broadcasting over LOCs, and we do initial work on the capacity region of constant-dimension multiplicative LOBCs (CMLOBCs), a generalization of broadcast erasure channels. Two fundamental problems regarding CMLOBCs are addressed-finding necessary and sufficient conditions for degradation and deciding whether time sharing suffices to achieve the boundary of the capacity region in the degraded case.

The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved. Based on this analysis, a general result, including all previous results as special cases, is established for the minimum distance of individual codes in a regular LDPC code ensemble.

This paper presents a cross-layer iterative decoder for irregular low-density parity-check (LDPC) codes which uses cyclic redundancy check (CRC) codes. The key idea of the decoder is to use correctly decoded frames as an aid for correcting the remaining erroneous frames. To accomplish this, the decoder exchanges the relevant information between layers by using the cross-layer design method and an iterative decoding architecture. Moreover, the unequal-error protection (UEP) property of irregular LDPC is exploited and both the multiple-error detection and single-error correction capabilities of the CRC code are used. Simulation results show that the proposed decoder outperforms the pure sum-product algorithm (SPA) decoder by a considerable gain while the increase in complexity is moderate. Furthermore, the error floor of irregular LDPC codes in the high Eb/NO regime can be lowered effectively. The proposed cross-layer iterative decoder can be used for any irregular LDPC coded wireless system to boost the performance and lower the error floor.

The statement given in the title is proved. Linear codes over chain rings (commutative and noncommutative) are a natural generalization of linear codes over finite fields and of linear codes over integer residue class rings of prime power order. In matters of linear representability there is no obvious reason why we should prefer one chain ring to the other. Yet, apart from Z/sub 4/, there is one further nontrivial chain ring with four elements: the ring Z/sub 2/[x]/(x/sup 2/) of dual numbers over Z/sub 2/. It is natural to ask about the linear representability of the Reed-Muller codes over this ring. For the sake of completeness, we reformulate here in an obvious way the definition of a linearly representable code.