Some aspects of Coding Theory

Abstract

The problem we would like to work on revolve primarily around the two areas, Algebraic newlineGeometry and Coding Theory. Linear error correcting codes associated to higher dimen newlinesional algebraic varieties deand#64257;ned over and#64257;nite and#64257;elds have been a topic of recent interest. Linear newlinecodes associated with Grassmann varieties, Schubert varieties have been studied extensively newline([36, 38, 41]). newlineIn this thesis, we reviewed the linear codes and their parameters such as length, dimension, newlinebounds, higher weights. LCD codes is a topic of recent interest to many researchers[3, 4, newline24, 42]. In this thesis, we have given new proof of Massey s theorem[21], which is one of the newlinemost important characterization of LCD codes. We also have given one new construction newlineof ternary LCD codes. Besides this, we constructed some ternary LCD codes attaining one newlinespeciand#64257;c kind of bound. We then focused on algebraic varieties and their connection with newlinelinear codes. Our main interest lies in the two projective varieties called Grassmann variety newlineand Schubert variety. We have concentrated our attention on error correcting capabilities newlineof Schubert codes. We explain the Grassmann and Schubert codes and the known results newlineof these codes. These codes have been studied by C. T. Ryan and K. M. Ryan [8], Nogin newline[12], and Ghorpade and Lachaud [38], Hansen, Johnsen, Ranestad [22], Ghorpade, Patil, newlinePillai [40]. We have found out generator matrices of Schubert codes in some special cases, newlinefurther computed Gr¨obner bases of ideals associated with such codes and their decoding newlineand error-correction in accordance with [15, 28, 29]. Finally, we propose our future plan of newlineresearch. newline newline