Mixed codes, which are error-correcting codes in the Cartesian product of different-sized spaces, model degrading storage systems well. While such codes have previously been studied for their algebraic properties (e.g., existence of perfect codes) or in the case of unbounded alphabet sizes, we focus on the case of finite alphabets, and generalize the Gilbert-Varshamov, sphere-packing, Elias-Bassalygo, and first linear programming bounds to that setting. In the latter case, our proof is also the first for the non-symmetric mono-alphabetic q-ary case using Navon and Samorodnitsky’s Fourier-analytic approach.

Abstract This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions that are able to mask u partially stuck cells while correcting at the same time t random errors. The process of “masking” determines a word whose entries coincide with writable levels at the (partially) stuck cells. For $$u>1$$ u > 1 and alphabet size $$q>2$$ q > 2 , our new constructions improve upon the required redundancy of known constructions for $$t=0$$ t = 0 , and require less redundancy for masking partially stuck cells than former works required for masking fully stuck cells (which cannot store any information). Second, we show that treating some of the partially stuck cells as erroneous cells can decrease the required redundancy for some parameters. Lastly, we derive Singleton-like, sphere-packing-like, and Gilbert–Varshamov-like bounds. Numerical comparisons state that our constructions match the Gilbert–Varshamov-like bounds for several code parameters, e.g., BCH codes that contain all-one word by our first construction.

Global Navigation Satellite System (GNSS) and New Radio (NR) signals based positioning are both specified for User Equipment (UE) positioning in a 3rd Generation Partnership Project (3GPP) network. They may also be fused to determine UE position by hybrid methods. In an urban scenario, the UE often suffers from non-line of sight (NLoS) propagation conditions from the satellite or the base station (BS). Identifying the links that have a line of sight (LoS) condition between the transmitter and the receiver is of paramount importance to enhance the accuracy of position estimates from GNSS, NR signals, or hybrids thereof. To address the issues with NLoS links, we propose a novel positioning solution fusing measurements made using GNSS and fifth-generation (5G) signals in the frequency range 1 (FR1) in an urban environment. We apply a supervised machine learning (ML) approach to classify LoS and NLoS for both GNSS and 5G signals based on the feature set. An extended Kalman filter (EKF) fuses observable measurements with LoS from both GNSS and 5G to estimate the UE position. We obtain positioning errors below 30 cm indoors, and below 2 m for 90% of all positioning fixes. Moreover, we observe that using our proposed fusion approach outperforms positioning using either NR signals or GNSS signals alone. We demonstrate that it is advantageous to deploy a transmission and reception point (TRP) at the areas where GNSS-based positioning shows degradations, as the results show that a single TRP for hybrid positioning already halves the positioning error compared to using only the LoS GNSS signals.

This work deals with error correction for nonvolatile memories that are partially defective at some levels. Such memory cells can only store incomplete information since some of their levels cannot be utilized entirely due to, e.g., wearout. On top of that, this paper corrects random errors t ≥ 1 that could happen among u partially defective cells while preserving their constraints. First, we show that the probability of violating the partially defective cells’ restriction due to random errors is not trivial. Next, we update the models in [1] such that the coefficients of the output encoded vector plus the error vector at the partially defective positions are non-zero. Lastly, we state a simple proposition (Proposition 3) for masking the partial defects using a code with a minimum distance d such that d ≥ 2(u + t) + 1. “Masking” means selecting a word whose entries correspond to writable levels in the (partially) defective positions. A comparison shows that masking u cells by this proposition for a particular BCH code is as effective as using the complicated coding scheme proven in [1, Theorem 1].