Projected subcodes of the second order binary Reed-Muller code

Let \(\mathcal R_{t,n}\) denote the set of t-resilient Boolean functions of n variables. First, we prove that the covering radius of the binary Reed-Muller code RM(2,6) in the sets \(\mathcal R_{t,6}\), t=0,1,2 is 16. Second, we show that the covering radius of the binary Reed-Muller code RM(2,7) in the set \(\mathcal R_{3,7}\) is 32. We derive a new lower bound for the covering radius of the Reed-Muller code RM(2,n) in the set \(\mathcal R_{n-1,4}\). Finally, we present new lower bounds in the sets \(\mathcal R_{t,7}\), t=0,1,2.

Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code C, identify for each dimension v, the smallest size of the support of a sub code of $C$ of dimension u, The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work [1] by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes. In the present paper, we derive a matching lower bound by adapting the proof techniques used originally for Reed-Muller (RM) codes by Wei in [2]. This results in a characterization of the GHW hierarchy of binary PRM codes.

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

A new class of binary codes is introduced. The codes have length n = 1.5 · 2m (m ≥ 3), and have a similar ‘double inductive’ structure to that possessed by the class of Reed-Muller (RM) codes. The new codes are structured such that their length-to-distance ratio n/d = 2m, m ≥ 2. For rates less than 1/2, the new codes have a higher rate than the next Reed-Muller code of the same order and length n = 2 · 2m.

Cycle indices of linear, affine, and projective groups

Recently introduced Polar coding is the first practical coding technique that can be proven to achieve the Shannon capacity for a multitude of communication channels. Polar codes are close to Reed-Muller codes except the fact that they are tuned for the parameters of the channel. Hence, Polar codes are shown to offer better performance, e.g., in the erasure channel. It is known that second order Reed-Muller codes can be used for Compressed Sensing. Inspired by that result, we propose Polar codes as measurement matrices in CS and compare their numerical performances. We also present the algebraic relation between the erasure channel and CS theory, and discuss fast solution techniques.

This paper presents an elementary method to independently obtain the weight distribution of second-order binary Reed-Muller codes RM(2, m). We note that such method can also be used to study the weight distribution of rth-order Reed-Muller codes for r ≥ 3.

We classify all $q$-ary $\\Delta$-divisible linear codes which are spanned by\ncodewords of weight $\\Delta$. The basic building blocks are the simplex codes,\nand for $q=2$ additionally the first order Reed-Muller codes and the parity\ncheck codes. This generalizes a result of Pless and Sloane, where the binary\nself-orthogonal codes spanned by codewords of weight $4$ have been classified,\nwhich is the case $q=2$ and $\\Delta=4$ of our classification. As an\napplication, we give an alternative proof of a theorem of Liu on binary\n$\\Delta$-divisible codes of length $4\\Delta$ in the projective case.\n

This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award. In this work, a flexible and robust private information retrieval (PIR) scheme based on binary non-maximum distance separable (non-MDS) codes is considered. This combines previous works on PIR schemes based on transitive non-MDS codes on one hand, and PIR from MDS-coded Byzantine and nonresponsive servers on the other hand. More specifically, a PIR scheme employing binary Reed–Muller (RM) codes tolerant to colluding, Byzantine, and non-responsive servers is constructed, and bounds for the achievable rates are derived under certain conditions. The construction of such schemes turns out to be much more involved than for MDS codes. Namely, the binary query vectors have to be selected with great care to hit the desired information sets, which is technically challenging as will be shown.

Enumeration of orbits on cycles for linear and affine groups

This paper introduces multidimensional generalizations of binary Reed-Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary Reed-Muller codes. The subspaces are associated with projection operators determined by Pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest. The Grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas, where separation with respect to the chordal distance on Grassmannian space guarantees closeness to the channel capacity. It is shown that the capacity of the noncoherent multiple-input-multiple-output (MIMO) channel at both low and moderate signal-to-noise ratio (SNR) (under the constraint that only isotropically distributed unitary matrices are used for information transmission) is closely approximated by these packings.

A multistage decoding algorithm is given for lattices obtained from a multilevel code formula. The algorithm is shown to have the same effective error-correcting radius as maximum-likelihood decoding, so that the performance loss is essentially determined by the increase in the effective error coefficient, for which an expression is given. The code formula generalizes some previous multilevel constructions to constructions of known single-level binary lattices with many levels, and then to decoders for them with the proposed algorithm. The trade-off between complexity reduction and performance loss is evaluated for several known lattices and two new ones, indicating that the approach is effective provided the binary codes involved in the code formula are not too short. All codes used in our constructions are binary Reed-Muller codes. >

An unsourced random access scheme using the second-order Reed-Muller (RM) sequences is proposed. Each of the active devices maps its message to an RM sequence and randomly selects a single subcarrier to transmit through a multipath fading channel. In a receiver, after OFDM demodulation, at each subcarrier the list of corresponding messages is blindly decoded by iterative RM decoding. Simulations show that the proposed systems provide low block error rates over multipath fading channels.

Restrictions on weight distribution of Reed-Muller codes

Locally testable codes (LTCs) of constant minimum (absolute) distance that allow the tester to make a nearly linear number of queries have become the focus of attention recently, due to their connections to central questions in approximability theory. In particular, the binary Reed-Muller code of block length N and absolute distance d is known to be testable with O(N/d) queries, and has a dimension of a N − (log N)log d. The polylogarithmically small co-dimension is the basis of constructions of small set expanders with many bad eigenvalues, and size-efficient PCPs based on a shorter version of the long code. The smallest possible co-dimension for a distance d code (without any testability requirement) is a d/2 log N, achieved by BCH codes. This raises the natural question of understanding where in the spectrum between the two classical families, Reed-Muller and BCH, the optimal co-dimension of a distance d LTC lies --- in other words the price one has to pay for local testability.One promising approach for constructing LTCs is to focus on affine-invariant codes, whose structure makes testing guarantees easier to deduce than for general codes. Along these lines, the authors of [HRZS13] and [GKS13] recently constructed an affine-invariant family of high-rate LTCs with slightly smaller co-dimension than Reed-Muller codes. In this work, we show that their construction is essentially optimal among linear affine-invariant LTCs that contain the Reed-Muller code of the appropriate degree.