The Golomb-Welch conjecture states that there are no perfect eerror-correcting codes in Z n for n ≥ 3 and e ≥ 2. In this note, we prove the nonexistence of perfect 2-error-correcting codes for a certain class of n, which is expected to be infinite. This result further substantiates the Golomb-Welch conjecture.

Since 1968, when the Golomb-Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb-Welch conjecture. Further, new results on Golomb-Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.

Several studies reported the effect of obesity on the progression of IgA nephropathy (IgAN). However, the impact of obesity on the clinicopathologic presentation of IgAN remains uncertain. This is a retrospective cross-sectional study from eight university hospitals in South Korea. Patients were categorized into three groups using the Asia-Pacific obesity classification based on body mass index (BMI). Clinical and histopathologic data at the time of renal biopsy were analyzed. Among 537 patients with IgAN, the obese group was more hypertensive and had lower estimated glomerular filtration rate and more proteinuria than other groups. The histologic scores for mesangial matrix expansion (MME), interstitial fibrosis, tubular atrophy, and mesangial C3 deposition differed significantly between the three groups. Among these histopathologic parameters, BMI was independently positively associated with MME score on multivariable linear regression analysis (p = 0.028). Using multivariable logistic regression analysis, the obese group was independently associated with higher MME scores compared to the normal weight/overweight group (p = 0.020). However, BMI was not independently associated with estimated glomerular filtration rate or proteinuria on multivariable analysis. Obesity was independently associated with severe MME in patients with IgAN. Obesity may play an important pathogenetic role in mesangial lesions seen in IgAN.

Objectives Contrast-enhanced ultrasound (CEUS) enables the assessment of real-time renal microcirculation. This study investigated CEUS-driven parameters as hemodynamic predictors for renal outcomes in patients with acute kidney injury (AKI). Methods Forty-eight patients who were diagnosed with AKI were prospectively enrolled and underwent CEUS at the occurrence of AKI. Parameters measured were the wash-in slope (WIS), time to peak intensity, peak intensity (PI), area under the time-intensity curve (AUC), mean transit time (MTT), time for full width at half maximum, and rise time (RT). The predictive performance of the CEUS-driven parameters for Kidney Disease Improving Global Outcomes (KDIGO) AKI stage, initiation of renal replacement therapy (RRT), AKI recovery, and chronic kidney disease (CKD) progression was assessed. Receiver operating characteristic (ROC) analysis was performed to evaluate the diagnostic performance of CEUS. Results Cortical RT (Odds ratio [OR] = 1.21) predicted the KDIGO stage 3 AKI. Cortical MTT (OR = 1.07) and RT (OR = 1.20) predicted the initiation of RRT. Cortical WIS (OR = 76.23) and medullary PI (OR = 1.25) predicted AKI recovery. Medullary PI (OR = 0.78) and AUC (OR = 1.00) predicted CKD progression. The areas under the ROC curves showed reasonable performance for predicting the initiation of RRT and AKI recovery. The sensitivity and specificity of the quantitative CEUS parameters were 60-83% and 62-77%, respectively, with an area under the curve of 0.69-0.75.