We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance d=4 , dimension k=3 of the codewords for all field sizes q, and sufficiently large dimensions v of the ambient space. It exceeds the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.
Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical bounds on the error probability based on the exact weight spectrum coefficients and the exact stopping size spectrum coefficients is presented. It is observed that decoding performance very close to the ML decoding performance can be achieved with a relatively small number of redundant rows for some codes, for both the BEC and the AWGN channels.
Locally repairable codes for distributed storage systems have gained a lot of interest recently, and various constructions can be found in the literature. However, most of the constructions result in either large field sizes and hence too high computational complexity for practical implementation, or in low rates translating into waste of the available storage space. In this paper we address this issue by developing theory towards code existence and design over a given field. This is done via exploiting recently established connections between linear locally repairable codes and matroids, and using matroid-theoretic characterisations of linearity over small fields. In particular, nonexistence can be shown by finding certain forbidden uniform minors within the lattice of cyclic flats. It is shown that the lattice of cyclic flats of binary matroids have additional structure that significantly restricts the possible locality properties of 𝔽_2 -linear storage codes. Moreover, a collection of criteria for detecting uniform minors from the lattice of cyclic flats of a given matroid is given, which is interesting in its own right.
The $\mathbb{Z}_{2^s}$-additive codes are subgroups of $\mathbb{Z}^n_{2^s}$, and can be seen as a generalization of linear codes over $\mathbb{Z}_2$ and $\mathbb{Z}_4$. A $\mathbb{Z}_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a $\mathbb{Z}_{2^s}$-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the $\mathbb{Z}_4$-linear Hadamard codes. In this paper, the kernel of $\mathbb{Z}_{2^s}$-linear Hadamard codes and its dimension are established for $s > 2$. Moreover, we prove that this invariant only provides a complete classification for some values of $t$ and $s$. The exact amount of nonequivalent such codes are given up to $t=11$ for any $s\geq 2$, by using also the rank and, in some cases, further computations.
This paper is devoted to giving a generalization from linear codes to the larger class of almost affine codes of two different results. One such result is how one can express the relative generalized Hamming weights of a pair of codes in terms of intersection properties between the smallest of these codes and subcodes of the largest code. The other result tells how one can find the extended weight polynomials, expressing the number of codewords of each possible weight, for each code in an infinite hierarchy of extensions of a code over a given alphabet. Our tools will be demi-matroids and matroids.
As affine-invariant codes, Reed-Muller codes are extension of cyclic group codes and they have a defining set that determines them uniquely. In this paper we identify those cyclic codes with multidimensional abelian codes and we use the techniques introduced in [3] to construct information sets for first and second order Reed-Muller codes from its defining set.
Random network coding is a method that achieves multicast capacity asymptotically for general networks [1, 7]. In this approach, vertices in the network randomly and linearly combine incoming information in a distributed manner before forwarding it through their outgoing edges. To ensure success, the involved finite field needs to be large enough [2, 7], which can be an obstacle if some inner (intermediate) nodes have less computational power than others. In this work, we analyze what can be achieved if different nodes are allowed to use different finite fields from a selection of fields all contained in some composite extension finite field [3, 5].
Transmission across asynchronous communication channels is subject to laser injection attacks which cause glitches, pulses that are added to the transmitted signal at arbitrary times, and delays. We present self-synchronizing coding schemes with low latency at the receiver that require no acknowledgement and can decode transmissions subject to random delays and distorted by random glitches.
We present a novel multidimensional network model as a means to analyze decoder failure and characterize trapping sets of graph-based codes. We identify a special class of these decoding networks, which we call transitive networks, and show how they may be used to identify trapping sets and inducing sets. Many codes have transitive decoding network representations. We conclude by investigating the decoding networks of codes arising from product, half-product, and protograph code constructions.
Linear codes that meet their dual trivially are also known as linear complementary dual codes. Quasi-abelian complementary dual codes are characterized using a known decomposition of a semisimple group algebra. Consequently, enumeration of such codes are obtained. More explicit formulas are given for the number of quasi-abelian complementary dual codes of index 2 with respect to Euclidean and Hermitian inner products. A sequence of asymptotically good binary quasi-abelian complementary dual codes of index 3 is constructed from an existing sequence of asymptotically good binary self-dual quasi-abelian codes of index 2.
We examine the presence of absorbing sets, fully absorbing sets, and elementary absorbing sets in low-density parity-check (LDPC) codes arising from certain classes of finite geometries. In particular, we analyze the absorbing set spectra of LDPC codes from finite Euclidean planes. For some parameters, we classify the absorbing sets present and give exact counts on their multiplicities.
We compare two popular tracing traitor schemes (1) using non-binary codes with identifiable parent property (IPP-codes) and (2) using family of sets with identifiable parent property. We establish a natural basis for comparing and show that the second approach is stronger than IPP-codes. We also establish a new lower bound on the cardinality of the family of sets with identifiable parent property.
Given a sequence of bits produced by a linear feedback shift register (LFSR), the Berlekamp-Massey algorithm finds a register of minimal length able to generate the sequence. The situation is different when the sequence is perturbed; for instance, when it is sent through a transmission channel. LFSRs can be described as autonomous systems. A perturbed sequence of bits generated by an LFSR can be interpreted as a codeword in the binary linear code generated by the corresponding observability matrix. The problem of finding the original sequence can then be stated as the decoding problem, "given the received codeword, find the information transmitted". We propose two decoding algorithms, one based on a brute force attack and the other one based on the representation technique of the syndromes introduced by Becker, Joux, May, and Meurer (2012).
In this paper we study the minimality of input-state-output (ISO) representations of basic two-dimensional (2D) convolutional codes. For that we consider the Fornasini-Marchesini ISO representations of such codes. We define the novel property of strongly modally reachable representations and we show that such representations are minimal representations of a basic 2D convolutional code. Moreover, we prove that the dimension of such minimal representations equals the complexity of the code.
Linear batch codes and codes for private information retrieval (PIR) with a query size t and a restricted size r of the reconstruction sets are studied. New bounds on the parameters of such codes are derived for small values of t or r by providing corresponding constructions. By building on the ideas of Cadambe and Mazumdar, a new bound in a recursive form is derived for batch codes and PIR codes.
In this paper we propose a woven block code construction based on two convolutional codes. We also propose a soft-input decoder that allows this construction to have better error correction performance than the turbo codes with a conventional decoder. Computer simulation has showed a 0.1 dB energy gain relative to the LTE turbo code. Asymptotically the proposed code has distance greater than the product of free distances of component codes.
In this paper we study periodically time-varying convolutional codes by means of input-state-output representations. Using these representations we investigate under which conditions a given time-invariant convolutional code can be transformed into an equivalent periodic time-varying one. The relation between these two classes of convolutional codes is studied for period 2. We illustrate the ideas presented in this paper by constructing a periodic time-varying convolutional code from a time-invariant one. The resulting periodic code has larger free distance than any time-invariant convolutional code with equivalent parameters.
The length function ℓ _q(r,R) is the smallest length of a q-ary linear code of covering radius R and codimension r. New upper bounds on ℓ _q(r,2) are obtained for odd r≥ 3 . In particular, using the one-to-one correspondence between linear codes of covering radius 2 and saturating sets in the projective planes over finite fields, we prove that ℓ _q(3,2)≤√(q(3ln q+lnln q))+√(q/3ln q)+3 and then obtain estimations of ℓ _q(r,2) for all odd r≥ 5 . The new upper bounds are smaller than the previously known ones. Also, the new bounds hold for all q, not necessary large, whereas the previously best known estimations are proved only for q large enough.
Let A_q(n,d) denote the maximum size of a q-ary code with size n and minimum distance d. For most values of n and d, only lower and upper bounds on A_q(n,d) are known. In this paper we present 19 new lower bounds where q ∈{3,4,5} . The bounds are based on codes whose automorphisms are prescribed by transitive permutation groups. An exhaustive computer search was carried out to find the new codes.
Robust codes are codes that can detect any nonzero error with nonzero probability. This property makes them useful in protecting hardware systems from fault injection attacks which cause arbitrary number of bit flips. There are very few high rate robust codes, non of them has minimum distance greater than two. Therefore, robust codes with error correction capability are derived by concatenation of linear codes with high rate robust codes. This paper presents a new construction of non-linear robust codes with error correction capability. The codes are built upon linear codes; however, the redundant symbols that were originally allocated to increase the minimum distance of the code, are modified to provide both correction capability and robustness. Consequently, the codes are more effective and have higher rate than concatenated codes of the same error masking probability.
