<p>This thesis shows how certain classes of binary constant weight codes can be represented geometrically using linear structures in Euclidean space. The geometric treatment is concerned mostly with codes with minimum distance 2(w - 1), that is, where any two codewords coincide in at most one entry; an algebraic generalization of parts of the theory also applies to some codes without this property. The presented theorems lead to a total of 18 improvements of the table of lower bounds on A(n,d,w) maintained by E. M. Rains and N. J. A. Sloane. Additional improvements have been made by finding new lexicographic codes.</p>

This thesis shows how certain classes of binary constant weight codes can be represented geometrically using linear structures in Euclidean space. The geometric treatment is concerned mostly with codes with minimum distance 2(w - 1), that is, where any two codewords coincide in at most one entry; an algebraic generalization of parts of the theory also applies to some codes without this property. The presented theorems lead to a total of 18 improvements of the table of lower bounds on A(n,d,w) maintained by E. M. Rains and N. J. A. Sloane. Additional improvements have been made by finding new lexicographic codes.

In this thesis some families of linear error correcting codes are presented. The reader will find a general description of binary codes and more specific details about linear codes such as Hamming, repetition codes, Reed-Muller codes, etc. To fully immerse ourselves in the methods of coding and decoding, we will introduce examples in order to contribute to the understanding of the theories.   In these times of much communication through computer technology, our daily lives involve a substantial amount of data transmission. It is essential that these data are transmitted without errors through the communication channels. Therefore, the scientific field of error-correcting codes holds a significant importance in many aspects of todays society.   The main goal of this thesis is to study linear block codes which belong to the class of binary codes. In this case we will attribute a more prominent role to first order Reed-Muller codes. / I den här uppsatsen kommer flera varianter av linjära felrättande koder att presenteras. Läsaren får ta del av en allmän beskrivning av binära koder och en mer detaljerad framställning av linjära koder så som Hamming, repetitionskod, Reed-Muller kod m.m. Tillsammans med en fördjupning i ämnet, avseende metoder för kodning och avkodning, kommer vi att ge exempel för att bidra till förståelsen.   Den digitala eran, som vi lever i, innefattar att datatransmission är en del av vår vardag. Vår frekventa användning av mobila enheter visar på hur viktigt det är att data överförs korrekt via kommunikationskanalerna. Av den anledningen är vetenskapen om felrättande koder högaktuell i dagens samhälle.   Det huvudsakliga syftet med uppsatsen är att studera linjära block-koder som tillhör klassen binära koder. I det här fallet kommer vi att fokusera lite extra på Reed-Muller koder av första ordningen.

In this thesis the relationship between Gröbner bases and algebraic coding theory is investigated, and especially applications towards linear codes, with Reed-Müller codes as an illustrative example. We prove that each linear code can be described as a binomial ideal of a polynomial ring, and that a systematic encoding algorithm for such codes is given by the remainder of the information word computed with respect to the reduced Gröbner basis. Finally we show how to apply the representation of a code by its corresponding polynomial ring ideal to construct a class of codes containing the so called primitive Reed-Müller codes, with a few examples of this result.