1,214 research outputs found
Stopping Set Distributions of Some Linear Codes
Stopping sets and stopping set distribution of an low-density parity-check
code are used to determine the performance of this code under iterative
decoding over a binary erasure channel (BEC). Let be a binary
linear code with parity-check matrix , where the rows of may be
dependent. A stopping set of with parity-check matrix is a subset
of column indices of such that the restriction of to does not
contain a row of weight one. The stopping set distribution
enumerates the number of stopping sets with size of with parity-check
matrix . Note that stopping sets and stopping set distribution are related
to the parity-check matrix of . Let be the parity-check matrix
of which is formed by all the non-zero codewords of its dual code
. A parity-check matrix is called BEC-optimal if
and has the smallest number of rows. On the
BEC, iterative decoder of with BEC-optimal parity-check matrix is an
optimal decoder with much lower decoding complexity than the exhaustive
decoder. In this paper, we study stopping sets, stopping set distributions and
BEC-optimal parity-check matrices of binary linear codes. Using finite geometry
in combinatorics, we obtain BEC-optimal parity-check matrices and then
determine the stopping set distributions for the Simplex codes, the Hamming
codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201
Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes
Locally repairable codes (LRCs) have emerged as an important coding scheme in
distributed storage systems (DSSs) with relatively low repair cost by accessing
fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs
have been widely investigated. Optimal LRCs via cyclic and constacyclic codes
provide significant benefit of elegant algebraic structure and efficient
encoding procedure. In this paper, we continue to consider the constructions of
optimal LRCs via cyclic and constacyclic codes with long code length.
Specifically, we first obtain two classes of -ary cyclic Singleton-optimal
-LRCs with length when and is
even, and length when and , respectively. To the best of our knowledge, this is the first
construction of -ary cyclic Singleton-optimal LRCs with length and
minimum distance . On the other hand, an LRC acheiving the
Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic
codes, we construct two new families of -ary perfect LRCs with length
, minimum distance and locality
Energy-Efficient Design of STAR-RIS Aided MIMO-NOMA Networks
Simultaneous transmission and reflection-reconfigurable intelligent surface
(STAR-RIS) can provide expanded coverage compared with the conventional
reflection-only RIS. This paper exploits the energy efficient potential of
STAR-RIS in a multiple-input and multiple-output (MIMO) enabled non-orthogonal
multiple access (NOMA) system. Specifically, we mainly focus on
energy-efficient resource allocation with MIMO technology in the STAR-RIS
assisted NOMA network. To maximize the system energy efficiency, we propose an
algorithm to optimize the transmit beamforming and the phases of the low-cost
passive elements on the STAR-RIS alternatively until the convergence.
Specifically, we first decompose the formulated energy efficiency problem into
beamforming and phase shift optimization problems. To efficiently address the
non-convex beamforming optimization problem, we exploit signal alignment and
zero-forcing precoding methods in each user pair to decompose MIMO-NOMA
channels into single-antenna NOMA channels. Then, the Dinkelbach approach and
dual decomposition are utilized to optimize the beamforming vectors. In order
to solve non-convex phase shift optimization problem, we propose a successive
convex approximation (SCA) based method to efficiently obtain the optimized
phase shift of STAR-RIS. Simulation results demonstrate that the proposed
algorithm with NOMA technology can yield superior energy efficiency performance
over the orthogonal multiple access (OMA) scheme and the random phase shift
scheme
Bounds and Constructions of Singleton-Optimal Locally Repairable Codes with Small Localities
Constructions of optimal locally repairable codes (LRCs) achieving
Singleton-type bound have been exhaustively investigated in recent years. In
this paper, we consider new bounds and constructions of Singleton-optimal LRCs
with minmum distance , locality and minimum distance and
locality , respectively. Firstly, we establish equivalent connections
between the existence of these two families of LRCs and the existence of some
subsets of lines in the projective space with certain properties. Then, we
employ the line-point incidence matrix and Johnson bounds for constant weight
codes to derive new improved bounds on the code length, which are tighter than
known results. Finally, by using some techniques of finite field and finite
geometry, we give some new constructions of Singleton-optimal LRCs, which have
larger length than previous ones
- β¦