Periodic Reporting for period 1 - DaRe (Data Reliability in Networks and Storage Memories) Reporting period: 2015-06-01 to 2017-05-31 Summary of the context and overall objectives of the project Data networks and storage media play a key role in our modern every-day life. However, these systems are highly susceptible to errors which makes successful reception/retrieval of the transmitted/stored data impossible. In data networks, errors occur due to interference with other users, due to multipath propagation of the signal, or due to component noise of the receiver. Data storage media like flash memories suffer from manufacturing imperfections, wearout, and fluctuating read/write errors. In order to transmit and store data reliably, high-performance error-correcting codes and efficient decoding algorithms are hence a necessary means. The objective of this project was to provide novel and superior approaches for reliability in data networks and storage media. By means of error-correcting codes and information-theoretic analyses, we made data transmission and storage more reliable against errors of different types. The first part of this project dealt with the construction of better and more efficient coding schemes for networks. In particular, we constructed subspace codes with high cardinality and low decoding complexity, and improved the reliability of multi-source networks. The second part of this project analysed the physical error model of non-volatile storage memories like flash memories and developed coding strategies which correct these errors and mask defect cells of the memory. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far 1.) Good Network Codes---By Ferrers Diagram Rank-Metric CodesIn random linear network coding (RLNC), each nodeforwards random linear combinations of all packets received sofar. The data packets can be seen as vectors over a finite field and the internal network structure is assumed tobe unknown. However, due to the linear combinations, onesingle erroneous packet can propagate widely throughout the whole network and can render the whole transmissionuseless. This makes error-correcting codes in RLNC essential to guarantee reliability. It was shown in by Kötter and Kschischang thatsubspace codes are highly suitable for this purpose. The multi-level construction by Etzion and Silberstein is one of the constructions providing codes with the largest known cardinalityfor subspace codes. This construction is based on the union of several lifted rank-metric codes,which are constructed in Ferrers diagrams. In this project, we have investigated and constructed optimal rank-metric codes in Ferrers diagrams.First, we have considered rank-metric anticodes and proved a code-anticode bound for Ferrers diagram rank-metric codes. Four techniques and constructions of Ferrers diagram rank-metriccodes were presented, each providing optimal codes for differentdiagrams and parameters for which no optimal solution was known before. 2.) Good Network Codes---Investigation of List Decodability of Gabidulin CodesSubspace codes can be applied for error-correction in network coding.A special class of almost-optimal subspace codes can be constructed by lifting rank-metric codes.Gabidulin codes can be seen as the rank-metricequivalent of Reed–Solomon codes. In this project, subspace codes were used to prove two boundson the list size in decoding certain Gabidulin codes. The firstbound is an existential one, showing that exponentially-sizedlists exist for codes with specific parameters. The second boundpresents exponentially-sized lists explicitly, for a different set ofparameters. Both bounds rule out the possibility of efficientlylist decoding several families of Gabidulin codes for any radiusbeyond half the minimum distance. Such a result was known sofar only for non-linear rank-metric codes, and not for Gabidulincodes. These results reveal a significant difference in list decodingGabidulin and Reed–Solomon codes, although the definitionsof these code classes strongly resemble each other. 3.) Codes for Partially Stuck-at Memory CellsIn this project, we have studied a new model of defectmemory cells, called partially stuck-at memory cells, which ismotivated by the behavior of multi-level cells in non-volatilememories such as flash memories and phase change memories.Our main contribution in the project is the study of codes formasking u partially stuck-at cells. We have first derived lower and upperbounds on the redundancy of such codes. We have then presented three codeconstructions over an alphabet of size q. Furthermore, we have studied the dual defect model inwhich cells cannot reach higher levels, and shown that codes forpartially stuck-at cells can be used to mask this type of defectsas well. Lastly, we have analyzed the capacity of the partially stuck-atmemory channel. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) 1.) Good Network Codes---By Ferrers Diagram Rank-Metric CodesBefore, only optimal codes for very special parameters were known.This work has already received significant attention at conferences and manyresearchers from other universities showed their interest in using thesecodes in applications. In future, we will investigate the impact on network coding of these constructions and perform simulations to demonstrate the performance gain. These results were also presented at conferences for the COST Action IC-1104,and therefore disseminated to the European network codingcommunity. Also, it is a collaboration of the Technion with the University ofNeuchatel in Switzerland and therefore a basis for a future collaborationbetween these two groups. Before this project, there was no collaborationbetween these two groups.2.) Good Network Codes---Investigation of List Decodability of Gabidulin CodesThis works proves that several classes of Gabidulin codes cannot belist decoded beyond the unique decoding radius.This was a fundamental open question in theoretical coding theoryand our results gained a lot of attention from famous researchers.We have presented these results at the IEEE International Symposiumon Information Theory 2015.Meanwhile this work was cited several times amongst others by Guruswami and his group who is one of the top researchers in coding theory and we have communicatedextensively with him and his group about our results.Therefore, these results have increased my international visibilitysignificantly and I have established new collaborations with renownresearchers and their groups.Further, based on my ideas, I have guided the PhD student Netanel Raviv to achieve these results. This has therefore contributed to my supervision skills.3.) Codes for Partially Stuck-at Memory CellsWithin in this project, together with Eitan Yaakobi, we were the firstresearchers who investigated the problem of partially stuck cellsfrom a coding point of view.It is therefore the first work which considers the practically quite relevant model of partially stuck cells and develops good (sometimes even optimal) code constructions. The significance of these code constructions was also already recognizedby industry. I have presented these results at SanDisk Israel, and they want to implement ourconstructions in hardware to see how good they perform.My talk and secondment and SanDisk have further contributedto establish a long-lasting collaboration between SanDisk andProf. Eitan Yaakobi at the Technion.This is a significant step in establishing research on memories inIsrael and Europe since so far, most of the research was done in the US. Comparison of the code of our constructions for masking partially stuck cells and the channel capaci Illustration of all Ferrers diagrams for which our subcode construction provides optimal codes. Illustration of all Ferrers diagrams for which our MDS construction provides optimal codes.