Asymptotically MDS Array BP-XOR Codes

Belief propagation or message passing on binary erasure channels (BEC) is a low complexity decoding algorithm that allows the recovery of message symbols based on bipartite graph prunning process. Recently, array XOR codes have attracted attention for storage systems due to their burst error recovery performance and easy arithmetic based on Exclusive OR (XOR)-only logic operations. Array BP-XOR codes are a subclass of array XOR codes that can be decoded using BP under BEC. Requiring the capability of BP-decodability in addition to Maximum Distance Separability (MDS) constraint on the code construction process is observed to put an upper bound on the maximum achievable code block length, which leads to the code construction process to become a harder problem. In this study, we introduce asymptotically MDS array BP-XOR codes that are alternative to exact MDS array BP-XOR codes to pave the way for easier code constructions while keeping the decoding complexity low with an asymptotically vanishing coding overhead. We finally provide and analyze a simple code construction method that is based on discrete geometry to fulfill the requirements of the class of asymptotically MDS array BP-XOR codes.Comment: 8 pages, 4 figures, to be submitte

Characteristics of study participants by status at the end of follow-up.

<p>Characteristics of study participants by status at the end of follow-up.</p

Hazard ratios (HR) and 95% confidence intervals (95% CI) for risk of death from BCa, overall cancer, cardiovascular disease and all causes by clinical cut-offs of CRP and tertiles of absolute granulocyte count and G/L ratio.

<p>All models were adjusted for age, race-ethnicity, cigarette smoking, alcohol consumption, vigorous physical activity, menopausal status, age of menarche, use of oral contraception and aspirin.</p

Hazard ratios (HR) and 95% confidence intervals (95% CI) for risk of death due to BCa, overall cancer, cardiovascular disease and all causes by continuous levels of clinically detectable CRP, absolute granulocyte count and G/L ratio, and trends from categories of markers, stratified by overweight status and median WC.

<p>All models were adjusted for age, race-ethnicity, cigarette smoking, alcohol consumption, vigorous physical activity, menopausal status, age of menarche, use of oral contraception and aspirin.</p

Cumulative mortality from BCa, other cancer and cardiovascular disease by the highest and lowest categories of CRP, AGC and G/L ratio.

<p>Gray’s test for equality of cumulative incidence functions was performed in presence of competing risks.</p

Hazard ratios for the risk of different types of cancer for groups of population based on tertiles of overall mean glucose and fructosamine.

<p>All models were adjusted for age, SES, fasting status, history of diabetes, lung and cardiovascular disease, serum albumin, total cholesterol and triglycerides. Additional adjustment for sex was performed for colorectal and lung cancer, as well as for parity and age at first childbirth for breast cancer. P-values for interaction were 0.29, 0.93, 0.01, and 0.08 for prostate, breast, colorectal and lung cancer, respectively.</p

Hazard ratios for overall cancer risk for standardized log fructosamine in different tertiles of glucose in fasting population.

<p>The model was adjusted for age, sex, SES, fasting status, history of diabetes, lung and cardiovascular disease, serum albumin, total cholesterol and triglycerides.</p

Hazard ratios and confidence intervals for the risk of overall and different types of cancer for standardized log overall mean glucose and fructosamine.

*<p>Interaction between glucose and fructosamine in relation to cancer risk.</p>1<p>Standardized log glucose and fructosamine were each analyzed in separated models; adjusted for age.</p>2<p>Adjusted for age, sex, SES, fasting status, history of diabetes, lung and cardiovascular disease, serum albumin, total cholesterol and triglycerides.</p>3<p>Subcohort of those with BMI values (N = 2,828).</p>4<p>Subcohort of nondiabetic persons, defined as those with serum glucose level <7.0 mmol/L at all measurements and without registered hospital discharge diagnosis of diabetes mellitus prior to the date of last measurement (N = 10,743); not adjusted for history of diabetes.</p>5<p>Subcohort of fasting persons; not adjusted for fasting status (N = 5,026);</p>6<p>Stratified analysis by glucose tertiles to evaluate the interaction between glucose and fructosamine; standardized log glucose was not included in the model.</p>7<p>Sex-stratified analysis in men; not adjusted for sex.</p>8<p>Sex-stratified analysis in women; not adjusted for sex; adjusted for parity and age at first childbirth.</p

Hazard ratios for overall cancer risk for groups of population based on tertiles of overall mean glucose and fructosamine.

<p>All models were adjusted for age, sex, SES, fasting status, history of diabetes, lung and cardiovascular disease, serum albumin, total cholesterol and triglycerides. P-value for interaction was 0.77.</p

Comparison between population with repeated measurements and single measurement of glucose and fructosamine in the AMORIS Study.

*<p>Standardized log glucose and fructosamine were each analyzed in the same models; adjusted for age, sex, SES, fasting status, history of diabetes, lung and cardiovascular disease, serum albumin, total cholesterol and triglycerides.</p