Non–existence of some 4–dimensional Griesmer codes over finite fields

We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$; $d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q \ge 9$, where $g_q(4,d)=\sum_{i=0}^{3} \left\lceil d/q^i \right\rceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for $2q^3-3q^2-3q+1 \le d \le 2q^3-3q^2$, $2q^3-5q^2-2q+1 \le d \le 2q^3-5q^2$ and $q^3-q^2-rq-2 \le d \le q^3-q^2-rq$ with $4 \le r \le 6$ for $q \ge 9$ and that $n_q(4,d) \ge g_q(4,d)+1$ for $2q^3-rq^2-2q+1 \le d \le 2q^3-rq^2-q$ for $3 \le r \le (q+1)/2$, $q \ge 5$ and $2q^3-4q^2-3q+1 \le d \le 2q^3-4q^2-2q$ for $q \ge 9$, where $n_q(4,d)$ denotes the minimum length $n$ for which an $[n,4,d]_q$ code exists

Non–existence of some 4–dimensional Griesmer codes over finite fields

We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 \le r \le (q+1)/2$, $q \ge 5$; $d=2q^3-3q^2-3q+1$ for $q \ge 9$; $d=2q^3-4q^2-3q+1$ for $q \ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q \ge 9$, where $g_q(4,d)=\sum_{i=0}^{3} \left\lceil d/q^i \right\rceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for $2q^3-3q^2-3q+1 \le d \le 2q^3-3q^2$, $2q^3-5q^2-2q+1 \le d \le 2q^3-5q^2$ and $q^3-q^2-rq-2 \le d \le q^3-q^2-rq$ with $4 \le r \le 6$ for $q \ge 9$ and that $n_q(4,d) \ge g_q(4,d)+1$ for $2q^3-rq^2-2q+1 \le d \le 2q^3-rq^2-q$ for $3 \le r \le (q+1)/2$, $q \ge 5$ and $2q^3-4q^2-3q+1 \le d \le 2q^3-4q^2-2q$ for $q \ge 9$, where $n_q(4,d)$ denotes the minimum length $n$ for which an $[n,4,d]_q$ code exists

Safety and Pitfalls of Blepharoptosis Surgery in Elderly People

Background Elderly patients often have complications of blepharoptosis surgery that can result in the appearance or exacerbation of superficial punctate keratopathy (SPK). However, postoperative changes to SPK status have not been previously reported. We used subjective assessment of symptoms and measurement of SPK scale classification to investigate the safety and efficacy of blepharoptosis surgery in elderly patients