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On representation of Hadamard Codes

Khalid H. Hameed --- Hameed k. Dawiod

Al-Fatih journal مجلة الفتح
ISSN: 87521996 Year: 2008 Volume: 4 Issue: 32 Pages: 102-109
Publisher: Diyala University جامعة ديالى

Abstract

Hadamard matrices (codes) were defined by the French mathematician M.J.Hadamard in 1893,[Hadamard," Resolution d’une question relative aux dêterminants" ,pp . 240-246,(1893)], called now Hadamard matrices.Hadamard matrix is a square array of +1,-1 whose rows and columns are mutually orthogonal.If the first row and first column contain only +1 ,the matrix is said to be in normal form.We can replace “+1” with “0” and “-1” with “1” to express Hadamard matrix using the logic elements {0,1}.
Since, Hadamard codes are orthogonal and belong to class of linear codes,they are used in error correcting codes.Error correcting codes (E.C.C)which are very useful in sending information over long distances or through channels where errors might occur in the message. Hadamard code was used by B.j.Falkowski and T.Sasao, [ Falkowski and Sasao, “Unified algorithm to generate Walash functions in four different orderings and its programmable hardware implementations”,p.822,2005], to generate Walsh functions.Walsh functions were invented in 1923 by the American mathematician J.L.Walsh (1895-1973) ,see,[ Walsh, “Aclosed set of normal orthogonal functions”,pp.5-24,(1923)]. Walsh functions are used in image processing, see [Yaroslavsky,”Digital holography and digital image processing: principle,methods,algorithms”,p.50].