An Adaptive Conditional Zero-Forcing Decoder with Full-diversity, Least Complexity and Essentially-ML Performance for STBCs

A low complexity, essentially-ML decoding technique for the Golden code and the 3 antenna Perfect code was introduced by Sirianunpiboon, Howard and Calderbank. Though no theoretical analysis of the decoder was given, the simulations showed that this decoding technique has almost maximum-likelihood (ML) performance. Inspired by this technique, in this paper we introduce two new low complexity decoders for Space-Time Block Codes (STBCs) - the Adaptive Conditional Zero-Forcing (ACZF) decoder and the ACZF decoder with successive interference cancellation (ACZF-SIC), which include as a special case the decoding technique of Sirianunpiboon et al. We show that both ACZF and ACZF-SIC decoders are capable of achieving full-diversity, and we give sufficient conditions for an STBC to give full-diversity with these decoders. We then show that the Golden code, the 3 and 4 antenna Perfect codes, the 3 antenna Threaded Algebraic Space-Time code and the 4 antenna rate 2 code of Srinath and Rajan are all full-diversity ACZF/ACZF-SIC decodable with complexity strictly less than that of their ML decoders. Simulations show that the proposed decoding method performs identical to ML decoding for all these five codes. These STBCs along with the proposed decoding algorithm outperform all known codes in terms of decoding complexity and error performance for 2,3 and 4 transmit antennas. We further provide a lower bound on the complexity of full-diversity ACZF/ACZF-SIC decoding. All the five codes listed above achieve this lower bound and hence are optimal in terms of minimizing the ACZF/ACZF-SIC decoding complexity. Both ACZF and ACZF-SIC decoders are amenable to sphere decoding implementation.Comment: 11 pages, 4 figures. Corrected a minor typographical erro

Full-Rate, Full-Diversity, Finite Feedback Space-Time Schemes with Minimum Feedback and Transmission Duration

In this paper a MIMO quasi static block fading channel with finite N-ary delay-free, noise-free feedback is considered. The transmitter uses a set of N Space-Time Block Codes (STBCs), one corresponding to each of the N possible feedback values, to encode and transmit information. The feedback function used at the receiver and the N component STBCs used at the transmitter together constitute a Finite Feedback Scheme (FFS). Although a number of FFSs are available in the literature that provably achieve full-diversity, there is no known universal criterion to determine whether a given arbitrary FFS achieves full-diversity or not. Further, all known full-diversity FFSs for T<N_t where N_t is the number of transmit antennas, have rate at the most 1. In this paper a universal necessary condition for any FFS to achieve full-diversity is given, using which the notion of Feedback-Transmission duration optimal (FT-Optimal) FFSs - schemes that use minimum amount of feedback N given the transmission duration T, and minimum transmission duration given the amount of feedback to achieve full-diversity - is introduced. When there is no feedback (N=1) an FT-optimal scheme consists of a single STBC with T=N_t, and the universal necessary condition reduces to the well known necessary and sufficient condition for an STBC to achieve full-diversity: every non-zero codeword difference matrix of the STBC must be of rank N_t. Also, a sufficient condition for full-diversity is given for the FFSs in which the component STBC with the largest minimum Euclidean distance is chosen. Using this sufficient condition full-rate (rate N_t) full-diversity FT-Optimal schemes are constructed for all (N_t,T,N) with NT=N_t. These are the first full-rate full-diversity FFSs reported in the literature for T<N_t. Simulation results show that the new schemes have the best error performance among all known FFSs.Comment: 12 pages, 5 figures, 1 tabl

Asymptotically-Optimal, Fast-Decodable, Full-Diversity STBCs

For a family/sequence of STBCs $\mathcal{C}_1,\mathcal{C}_2,\dots$, with increasing number of transmit antennas $N_i$, with rates $R_i$ complex symbols per channel use (cspcu), the asymptotic normalized rate is defined as $\lim_{i \to \infty}{\frac{R_i}{N_i}}$. A family of STBCs is said to be asymptotically-good if the asymptotic normalized rate is non-zero, i.e., when the rate scales as a non-zero fraction of the number of transmit antennas, and the family of STBCs is said to be asymptotically-optimal if the asymptotic normalized rate is 1, which is the maximum possible value. In this paper, we construct a new class of full-diversity STBCs that have the least ML decoding complexity among all known codes for any number of transmit antennas $N>1$ and rates $R>1$ cspcu. For a large set of $\left(R,N\right)$ pairs, the new codes have lower ML decoding complexity than the codes already available in the literature. Among the new codes, the class of full-rate codes ($R=N$) are asymptotically-optimal and fast-decodable, and for $N>5$ have lower ML decoding complexity than all other families of asymptotically-optimal, fast-decodable, full-diversity STBCs available in the literature. The construction of the new STBCs is facilitated by the following further contributions of this paper:(i) For $g > 1$, we construct $g$-group ML-decodable codes with rates greater than one cspcu. These codes are asymptotically-good too. For $g>2$, these are the first instances of $g$-group ML-decodable codes with rates greater than $1$ cspcu presented in the literature. (ii) We construct a new class of fast-group-decodable codes for all even number of transmit antennas and rates $1 < R \leq 5/4$.(iii) Given a design with full-rank linear dispersion matrices, we show that a full-diversity STBC can be constructed from this design by encoding the real symbols independently using only regular PAM constellations.Comment: 16 pages, 3 tables. The title has been changed.The class of asymptotically-good multigroup ML decodable codes has been extended to a broader class of number of antennas. New fast-group-decodable codes and asymptotically-optimal, fast-decodable codes have been include

Coded Data Rebalancing: Fundamental Limits and Constructions

Distributed databases often suffer unequal distribution of data among storage nodes, which is known as `data skew'. Data skew arises from a number of causes such as removal of existing storage nodes and addition of new empty nodes to the database. Data skew leads to performance degradations and necessitates `rebalancing' at regular intervals to reduce the amount of skew. We define an r-balanced distributed database as a distributed database in which the storage across the nodes has uniform size, and each bit of the data is replicated in r distinct storage nodes. We consider the problem of designing such balanced databases along with associated rebalancing schemes which maintain the r-balanced property under node removal and addition operations. We present a class of r-balanced databases (parameterized by the number of storage nodes) which have the property of structural invariance, i.e., the databases designed for different number of storage nodes have the same structure. For this class of r-balanced databases, we present rebalancing schemes which use coded transmissions between storage nodes, and characterize their communication loads under node addition and removal. We show that the communication cost incurred to rebalance our distributed database for node addition and removal is optimal, i.e., it achieves the minimum possible cost among all possible balanced distributed databases and rebalancing schemes

On Locally Decodable Index Codes

Index coding achieves bandwidth savings by jointly encoding the messages demanded by all the clients in a broadcast channel. The encoding is performed in such a way that each client can retrieve its demanded message from its side information and the broadcast codeword. In general, in order to decode its demanded message symbol, a receiver may have to observe the entire transmitted codeword. Querying or downloading the codeword symbols might involve costs to a client -- such as network utilization costs and storage requirements for the queried symbols to perform decoding. In traditional index coding solutions, this 'client aware' perspective is not considered during code design. As a result, for these codes, the number of codeword symbols queried by a client per decoded message symbol, which we refer to as 'locality', could be large. In this paper, considering locality as a cost parameter, we view index coding as a trade-off between the achievable broadcast rate (codeword length normalized by the message length) and locality, where the objective is to minimize the broadcast rate for a given value of locality and vice versa. We show that the smallest possible locality for any index coding problem is 1, and that the optimal index coding solution with locality 1 is the coding scheme based on fractional coloring of the interference graph. We propose index coding schemes with small locality by covering the side information graph using acyclic subgraphs and subgraphs with small minrank. We also show how locality can be accounted for in conventional partition multicast and cycle covering solutions to index coding. Finally, applying these new techniques, we characterize the locality-broadcast rate trade-off of the index coding problem whose side information graph is the directed 3-cycle.Comment: 10 pages, 1 figur

Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as Cn(r,m), and their duals are denoted as Bn(r,m). The length of these codes is nm, where n ≥ 2, and r is their 'order'. When n = 2, Cn(r,m) is the RM code of order r and length 2m. The special case of these codes corresponding to n being an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer to Bn(r,m) as the Berman code and Cn(r,m) as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group. Applying a result of Kumar et al. (2016) to this set of automorphisms, we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding. © 2022 IEEE