Abstract-This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.

Soft decision decoding of linear block codes based on ordered statistics is discussed in the context of multilevel coding with multistage decoding. Analytical expressions on the ordered statistics and the associated bit error probability have been derived. The result shows tradeos between error performance and decoding complexity in multilevel signaling including BPSK modulation formats as a special case.

This study is to investigate the clinical characteristics of late pregnancy with asymptomatic 2019 novel coronavirus disease (COVID-19) infection, evaluate the outcome of maternal and fetal prognosis, and identify the evidence of intrauterine vertical transmission. A 22-years-old pregnant woman with asymptomatic COVID-19 infection who was admitted to our hospital on 11 February 2020 was enrolled in this study. Clinical data including laboratory test results and chest computed tomography (CT) scanning were collected and reviewed. Diagnosis of late pregnancy with asymptomatic COVID-19 infection was made. Lumbar anesthesia for cesarean section was performed and a female baby was delivered uneventfully, with the Apgar score of 9 to 10 points. Three times of COVID-19 nucleic acid test for the baby was negative after delivery. The puerpera returned to normal after the operation and two times of throat swab COVID-19 nucleic acid test were all negative after antiviral therapy. We reported an asymptomatic COVID-19 pregnant woman with detailed clinical information and our result indicated that for late pregnant women with asymptomatic COVID-19 infection, there might be no intrauterine infection caused by vertical transmission. K E Y W O R D Sasymptomatic COVID-19 infection, intrauterine infection, late pregnancy, vertical transmission

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Abstract-This letter presents an algebraic method for constructing regular low-density parity-check (LDPC) codes based on Reed-Solomon codes with two information symbols. The construction method results in a class of LDPC codes in Gallager's original form. Codes in this class are free of cycles of length 4 in their Tanner graphs and have good minimum distances. They perform well with iterative decoding.

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