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SUMMARY:Simeon Ball (UPC)
DTSTART:20210325T153000Z
DTEND:20210325T163000Z
DTSTAMP:20260423T024720Z
UID:UCDANT/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCDANT/12/">
 Additive codes over finite fields</a>\nby Simeon Ball (UPC) as part of Dub
 lin Algebra and Number Theory Seminar\n\n\nAbstract\nIf $A$ is an abelian 
 group then we define an additive code to be a code $C$ with the property t
 hat for all $u\,v \\in C$\, we have $u+v \\in C$. If $A$ is a finite field
  then $C$ is linear over some subfield of $A$\, so we take $A={\\mathbb F}
 _{q^h}$ and assume that $C$ is linear over ${\\mathbb F}_q$.\n\nI will spe
 nd the first part of the talk talking about the geometry of linear\, addit
 ive and quantum stabiliser codes. \n\nThe second part of the talk (joint w
 ork with Michel Lavrauw and Guillermo Gamboa) will concern additive MDS co
 des. An {\\em MDS code} $C$ is a subset of $A^n$ of size $|A|^k$ in which 
 any two elements of $C$ differ in at least $n-k+1$ coordinates. In other w
 ords\, the minimum (Hamming) distance $d$ between any two elements of $C$ 
 is $n-k+1$. \n\n\n\nThe trivial upper bound on the length $n$ of a $k$-dim
 ensional additive MDS code over ${\\mathbb F}_{q^h}$ is\n$$\nn \\leqslant 
 q^h+k-1.\n$$\n\n\nThe classical example of an MDS code is the Reed-Solomon
  code\, which is the evaluation code of all polynomials of degree at most 
 $k-1$ over ${\\mathbb F}_{q^h}$. The Reed-Solomon code is linear over ${\\
 mathbb F}_{q^h}$ and has length $q^h+1$.\n\nThe MDS conjecture states (exc
 epting two specific cases) that an MDS code has length at most $q^h+1$. In
  other words\, there are no better MDS codes than the Reed-Solomon codes.\
 n\nWe use geometrical and computational techniques to classify all additiv
 e MDS codes over ${\\mathbb F}_{q^h}$ for $q^h \\in \\{4\,8\,9\\}$. We als
 o classify the longest additive MDS codes over ${\\mathbb F}_{16}$ which a
 re linear over ${\\mathbb F}_4$. These classifications not only verify the
  MDS conjecture for additive codes in these cases but also confirm there a
 re no additive non-linear MDS codes that perform as well as their linear c
 ounterparts. \n\nIn this talk\, I will cover the main geometrical theorem 
 that allows us to obtain this classification and compare these classificat
 ions with the classifications of {\\bf all} MDS codes of alphabets of size
  at most $8$\, obtained previously by Alderson (2006)\, Kokkala\, Krotov a
 nd Östergård (2015) and Kokkala and Östergård (2016).\n
LOCATION:https://researchseminars.org/talk/UCDANT/12/
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