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The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction.

Hormone receptor status is determined primarily to identify breast cancer patients who may benefit from hormonal therapy. The current clinical practice for the testing using either Allred score or H-score is still based on laborious manual counting and estimation of the amount and intensity of positively stained cancer cells in immunohistochemistry (IHC)-stained slides. This work integrates cell detection and classification workflow for breast carcinoma estrogen receptor (ER)-IHC-stained images and presents an automated evaluation system. The system first detects all cells within the specific regions and classifies them into negatively, weakly, moderately, and strongly stained, followed by Allred scoring for ER status evaluation. The generated Allred score relies heavily on accurate cell detection and classification and is compared against pathologists’ manual estimation. Experiments on 40 whole-slide images show 82.5% agreement on hormonal treatment recommendation, which we believe could be further improved with an advanced learning model and enhancement to address the cases with 0% ER status. This promising system can automate the exhaustive exercise to provide fast and reliable assistance to pathologists and medical personnel. The system has the potential to improve the overall standards of prognostic reporting for cancer patients, benefiting pathologists, patients, and also the public at large.