The Ohio University-Ohio State University Ring Theory Seminar presents a double feature on Friday, April 7, from 4:30 to 6:15 p.m. in Columbus.
The first speaker is Asiyeh Rafieipour from the University of Kashan (Kashan, Iran) discussing “On MDS and Near-MDS Codes” from 4:30-5:15 p.m. in Cockins Hall 240, OSU-Columbus.
Abstract: From the Singleton bound it follows that an $(n, k, d)_{q}$ code satisfies $d \leq n−k+1$. The Singleton defect of an $(n, k, d)_{q}$ code is $S(C) = n−k +1−d$. An $(n, k, d)_{q}$ code $C$ with $S(C) = 0$ is called a maximum distance separable (MDS) code which was defined by R.C. Singleton in 1964. The dual code of an MDS code is MDS. Codes of singleton defect 1 are called Almost-MDS (AMDS) codes. Near-MDS(NMDS) codes have been defined by S. Dodunekov and I. Landgev, in 1995, as the ‘next to the best’ class of codes by weakening some restrictions in the definition of MDS codes. NMDS codes have similar properties to MDS codes. Some celebrated non-binary codes, such as the ternary Golay codes, the quaternary quadratic residue [11,6,5] and the quaternary extended quadratic residue [12, 6, 6] codes, turn out to be near MDS codes. Unfortunately, unlike MDS codes, we have no nice infinite class of NMDS codes. In this talk, we introduce these codes and review some works on them and then we construct a family of MDS and NMDS codes.
The second speaker is Javier Ronquillo of Ohio University discussing “Urysohn Space and Distance Matrices” from 5:25 to 6:15 p.m. in CH 240.
Abstract: In 1927 Urysohn proved the existence of a separable complete metric space $\mathbb{U}$ that has the following properties:
1) (Universality) For each separable complete metrizable topological space $X$ there exist and isometric embedding of $X$ into $\mathbb{U}$
2) (Ultrahomogeneity) For any $f:A \rightarrow B$ isometry between finite subsets A, B of $\mathbb{U}$ there exists an extension of $f$ from $\mathbb{U}$ onto itself.
3) (Uniqueness) $mathbb{U}$ is unique up to isometries.
Since then there have been different constructions of this space. In 2002 A.M. Vershik studied the cone of distance matrices, these are matrices where the $i,j$ entry is $\rho(x_i,x_j)$ where $\rho$ is a pseudo-metric on an ordered countable set $\{x_1, x_2,…\}$. Using topological and geometric properties of this cone of distance matrices Vershik proved the existence of infinite universal matrices which is equivalent to the existence of $\mathbb{U}$. In this talk we discuss some properties of the Urysohn space $mathbb{U}$ and explore the techniques used by Vershik to prove its existence.
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