Artin-Schreier families and 2-D cycle codes

We start with the study of certain Artin-Schreier families. Using coding theory techniques, we determine a necessary and sufficient condition for such families to have a nontrivial curve with the maximum possible number of rational points over the finite field in consideration. This result produces several nice corollaries, including the existence of certain maximal curves; i.e., curves meeting the Hasse-Weil bound.We then present a way to represent two-dimensional (2-D) cyclic codes as trace codes starting from a basic zero set of its dual code. This representation enables us to relate the weight of a codeword to the number of rational points on certain Artin-Schreier curves via the additive form of Hilbert’s Theorem 90. We use our results on Artin-Schreier families to give a minimum distance bound for a large class of 2-D cyclic codes. Then, we look at some specific classes of 2-D cyclic codes that are not covered by our general result. In one case, we obtain the complete weight enumerator and show that these types of codes have two nonzero weights. In the other cases, we again give minimum distance bounds. We present examples, in some of which our estimates are fairly effcient

Linear Complementary Pair Of Group Codes over Finite Chain Rings

Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side-channel and fault injection attacks. The security parameter for an LCP of codes $(C,D)$ is defined as the minimum of the minimum distances $d(C)$ and $d(D^\bot)$. It has been recently shown that if $C$ and $D$ are both 2-sided group codes over a finite field, then $C$ and $D^\bot$ are permutation equivalent. Hence the security parameter for an LCP of 2-sided group codes $(C,D)$ is simply $d(C)$. We extend this result to 2-sided group codes over finite chain rings

Use of infiltrative anesthesia in acute anterior dislocation of shoulder

BACKGROUND: To evaluate the results of infiltrative anesthesia for manual closed reduction of acute primary anterior shoulder dislocation.   MATERIAL AND METHODS: A total of 55 adults with acute anterior dislocation of shoulder who were treated with Hippocratic maneuver were evaluated. Infiltrative anesthesia was applied directly to the deltoid muscle from two anatomic locations in anterolateral and posterolateral of the shoulder with prilocaine hydrochloride and bupivacaine (Citanest® + Marcaine®) was applied to all patients. All patients’ reductions were made by the same orthopaedic surgeon. Visual Analog Scale (VAS) of pain was applied to all subjects for evaluating the pain in management after the treatment. Demographic and clinical data, time of duration for reduction, and duration of hospitalization were recorded.   RESULTS: Mean age was 57.9 ± 4.5 years, 22% were women. The reduction was completed with the mean duration of 1.0 ± 0.3 minutes after applying infiltrative anesthesia. The mean VAS scores of the patients used infiltrative anesthesia were 4.6 which indicated moderate pain. The treatment was completed in the emergency room so that patients could be discharged after reduction in the emergency department. No recurrence and complications were observed in the one-year follow-up period.   CONCLUSION: Our study showed that infiltrative anesthesia, in addition to its easy management by orthopaedic surgeons, allows successful and fast reduction by avoiding difficulties caused by the contraction of the deltoid muscle without necessitating sedoanalgesia or general anesthesia

Artin-Schreier curves and weights of two dimensional cyclic codes

Let GF(q) be the finite field with q elements of characteristic p, GF(q^m) be the extension of degree m>1 and f(x) be a polynomial over GF(q^m). We determine a necessary and sufficient condition for y^q-y=f(x) to have the maximum number of affine GF(qm)-rational points. Then we study the weights of 2-D cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin-Schreier curves.We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds

A Bound on the number of rational points of certain Artin-Schreier families

Let F[sub q] = F[sub p]l for some 1 > 0 and consider the extension F[sub q][sup m] with m > 1. We consider families of curves of the form F = {y[sup q] - y = λ[sub 1]x[sup i1]+ λ[sub 2]x[sup i2] + ··· + λ[sub s]x[sup is]; λ[sub j] ∈ F[sub q]m, i[sub j] > 0 }. We call such families Artin-Schreier families, even though not every curve in F need be an Artin-Schreier curve. It is easy to see that the members of such a family can have at most q[sup m+l] affine F[sub q[sup m]]-rational points. Using a well-known coding theory technique, we determine the condition under which F can attain this bound and we obtain some simple, but interesting, corollaries of this result. One of these consequences shows the existence of maximal curves of Artin-Schreier type. Our main result is important for minimum distance analysis of certain two-dimensional cyclic code

ARTIN-SCHREIER FAMILIES AND 2-D CYCLIC CODES

I was very lucky to have Robert F. Lax as my advisor. He suggested a dissertation topic which has been quite pleasant and interesting to work on. He has been a great source of guidance and support for my work, and I thank him for that. I should also acknowledge the fine research environment at the Mathematics Department of Louisiana State University. Helpful comments provided by Arnaldo Garcia on parts of my work are also appreciated. I must especially thank my wife, Ceren, who has been very patient and support-ive during my studies. She has been a great companion. Finally, we both would like to thank our families for all the things that they did for our happiness and the support that they provided from very far away

Supersingular curves over finite fields and weight divisibility of codes

Motivated by a recent article of the second author, we relate a family of Artin-Schreier type curves to a sequence of codes. We describe the algebraic structure of these codes, and we show that they are quasi-cyclic codes. We show that if the family of Artin-Schreier type curves consists of supersingular curves then the weights in the related codes are divisible by a certain power of the characteristic. We give some applications of the divisibility result, including showing that some weights in certain cyclic codes are eliminated in subcodes

Supersingular curves over finite fields and weight divisibility of codes

Motivated by a recent article of the second author, we relate a family of Artin-Schreier type curves to a sequence of codes. We describe the algebraic structure of these codes, and we show that they are quasi-cyclic codes. We show that if the family of Artin-Schreier type curves consists of supersingular curves then the weights in the related codes are divisible by a certain power of the characteristic. We give some applications of the divisibility result, including showing that some weights in certain cyclic codes are eliminated in subcodes

The Concatenated Structure of Quasi-Cyclic Codes and an Improvement of Jensen's Bound

Following Jensen's work from 1985, a quasi-cyclic code can be written as a direct sum of concatenated codes, where the inner codes are minimal cyclic codes and the outer codes are linear codes. We observe that the outer codes are nothing but the constituents of the quasi-cyclic code in the sense of Ling-Sole. This concatenated structure enables us to recover some earlier results on quasi-cyclic codes in a simple way, including one of our recent results which says that a quasi-cyclic code with cyclic constituent codes are 2-D cyclic codes. In fact, we obtain a generalization of this result to multidimensional cyclic codes. The concatenated structure also yields a lower bound on the minimum distance of quasi-cyclic codes, as noted by Jensen, which we call Jensen's bound. We show that a recent lower bound on the minimum distance of quasi-cyclic codes that we obtained is in general better than Jensen's lower bound