Alexey A. Frolov

Bounds on the Parameters of Locally Recoverable Codes

A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert-Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets.

Multiple access system for a vector disjunctive channel

We address the problem of constructing a multiple access system for a disjunctive vector channel, similar to a multiuser channel without intensity information as described in [1]. To solve the problem, a signal-code construction based on nonbinary codes is proposed. For the resulting multiple access system, a lower bound on the relative group rate is derived. The bound coincides asymptotically with an upper bound.

Asymptotic estimation of the fraction of errors correctable by q-ary LDPC codes

We consider an ensemble of random q-ary LDPC codes. As constituent codes, we use q-ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.

On a multiple-access in a vector disjunctive channel

We address the problem of increasing the sum rate in a multiple-access system from [1] for small number of users. We suggest an improved signal-code construction in which in case of a small number of users we give more resources to them. For the resulting multiple-access system a lower bound on the relative sum rate is derived. It is shown to be very close to the maximal value of relative sum rate in [1] even for small number of users. The bound is obtained for the case of decoding by exhaustive search. We also suggest reduced-complexity decoding and compare the maximal number of users in this case and in case of decoding by exhaustive search.

On the capacity of a multiple-access vector adder channel

We investigate the capacity of the Q-frequency S-user vector adder channel (channel with intensity information) introduced by Chang and Wolf. Both coordinated and uncoordinated types of transmission are considered. Asymptotic (under the conditions Q → ∞, S = γQ, 0 < γ < ∞) upper and lower bounds on the relative (per subchannel) capacity are derived. The lower bound for the coordinated case is shown to increase with γ. At the same time, the relative capacity for the uncoordinated case is upper bounded by a constant.

A new coding method for a multiple-access system with a large number of active users

The paper addresses the problem of constructing an asynchronous multiple access system for a multi-user Q-frequency channel with additive white Gaussian noise (AWGN). We propose a coding scheme for the channel which allows a large number of users to work simultaneously in the system. The scheme combines the ideas of Frequency Shift Keying (FSK) and Frequency Hopping Spread Spectrum (FHSS). The major component of the scheme is a non-binary low-density parity-check (LDPC) code. The efficiency of the resulting multiple-access system is shown by simulations.

A coding technique for Q -frequency S -user gaussian channel

The paper addresses the problem of constructing an asynchronous multiple access system for a multiuser Q-frequency channel with additive white gaussian noise (AWGN). To solve the problem we propose a coding scheme for the channel. The major component of the scheme is non-binary low-density parity-check (LDPC) code. To increase the transmission rate we introduce the embedded modulation. The efficiency of the resulting multiple-access system is shown by simulations.

Bounds on the minimum code distance for nonbinary codes based on bipartite graphs

The minimum distance of codes on bipartite graphs (BG codes) over GF(q) is studied. A new upper bound on the minimum distance of BG codes is derived. The bound is shown to lie below the Gilbert-Varshamov bound when q ≤ 32. Since the codes based on bipartite expander graphs (BEG codes) are a special case of BG codes and the resulting bound is valid for any BG code, it is also valid for BEG codes. Thus, nonbinary (q ≤ 32) BG codes are worse than the best known linear codes. This is the key result of the work. We also obtain a lower bound on the minimum distance of BG codes with a Reed-Solomon constituent code and a lower bound on the minimum distance of low-density parity-check (LDPC) codes with a Reed-Solomon constituent code. The bound for LDPC codes is very close to the Gilbert-Varshamov bound and lies above the upper bound for BG codes.

An upper bound on the minimum distance of LDPC codes over GF(q)

In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over GF(q). The comparison to the lower bound for LDPC codes over GF(q) and to the upper bound for non-binary codes is done. The new bound is shown to lie under the Gilbert-Varshamov bound at high rates.

On the capacity of a PPM UWB multiple-access system with a single user noncoherent reception

We consider an uncoordinated multiple-access system that employs a modulation technique, in which the probability of suppressing the signal sent by a certain user can be considered negligible, to transmit information via a wireless channel (e.g. time hopping (TH) with pulse position modulation (PPM)).This channel can be considered as an A channel (channel without intensity information) [1]. For this channel a new method of transmission is proposed. The expression for the capacity of a multiple access system employing the proposed transmission method (for the single user reception case) is obtained. Both non-asymptotic and asymptotic formulas are derived and asymptotic behavior of the capacity (for the single user reception case) is studied.