NDLI: High-Girth matrices and polarization

Content Provider	IEEE Xplore Digital Library
Author	Abbe, E. Wigderson, Y.
Copyright Year	2015
Description	Author affiliation: Dept. of Math., Princeton Univ., Princeton, NJ, USA (Wigderson, Y.) \|\| Prog. in Appl. & Comput. Math. & EE Dept., Princeton Univ., Princeton, NJ, USA (Abbe, E.)
Abstract	The girth of a matrix is the least number of linearly dependent columns, in contrast to the rank which is the largest number of linearly independent columns. This paper considers the construction of high-girth matrices, whose probabilistic girth is close to their rank. Random matrices can be used to show the existence of high-girth matrices. This paper uses a recursive construction based on conditional ranks (inspired by polar codes) to obtain a deterministic and efficient construction of high-girth matrices for arbitrary relative ranks. Interestingly, the construction is agnostic to the underlying field and applies to both finite and continuous fields with the same binary matrix. The construction gives in particular the following: (i) over the binary field, high-girth matrices are equivalent to capacity-achieving codes, and our construction turns out to match exactly the BEC polar codes (even at finite block length). It hence gives a different interpretation of BEC polar codes, using the parity-check matrix instead of the generator matrix, and basic linear algebra instead of the mutual information, and generalizes to larger fields; (ii) for the BSC, our construction gives an operational meaning to the Bhattacharyya upper-bound process used in polar codes; (iii) for the reals, it gives an explicit candidate matrix for sparse recovery.
Starting Page	2461
Ending Page	2465
File Size	247294
Page Count	5
File Format	PDF
ISSN	21578117
e-ISBN	9781467377041
DOI	10.1109/ISIT.2015.7282898
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-06-14
Publisher Place	China
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Probabilistic logic Sparse matrices Linear codes Decoding Mutual information Parity check codes
Content Type	Text
Resource Type	Article

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