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Parity codes

Paulo E. D. Pinto, Fábio Protti, Jayme L. Szwarcfiter (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Motivated by a problem posed by Hamming in 1980, we define even codes. They are Huffman type prefix codes with the additional property of being able to detect the occurrence of an odd number of 1-bit errors in the message. We characterize optimal even codes and describe a simple method for constructing the optimal codes. Further, we compare optimal even codes with Huffman codes for equal frequencies. We show that the maximum encoding in an optimal even code is at most two bits larger than the maximum...

Parity codes

Paulo E. D. Pinto, Fábio Protti, Jayme L. Szwarcfiter (2010)

RAIRO - Theoretical Informatics and Applications

Motivated by a problem posed by Hamming in 1980, we define even codes. They are Huffman type prefix codes with the additional property of being able to detect the occurrence of an odd number of 1-bit errors in the message. We characterize optimal even codes and describe a simple method for constructing the optimal codes. Further, we compare optimal even codes with Huffman codes for equal frequencies. We show that the maximum encoding in an optimal even code is at most two bits larger than the maximum...

Picture codes

Symeon Bozapalidis, Archontia Grammatikopoulou (2006)

RAIRO - Theoretical Informatics and Applications

We introduce doubly-ranked (DR) monoids in order to study picture codes. We show that a DR-monoid is free iff it is pictorially stable. This allows us to associate with a set C of pictures a picture code B(C) which is the basis of the least DR-monoid including C. A weak version of the defect theorem for pictures is established. A characterization of picture codes through picture series is also given.

Poids des duaux des codes BCH de distance prescrite 2 a + 1 et sommes exponentielles

Éric Férard (2002)

Bulletin de la Société Mathématique de France

Soit n un entier pair. On considère un code BCH binaire C n de longueur 2 n - 1 et de distance prescrite 2 a + 1 avec a ≥ 3 . Le poids d’un mot non nul du dual de  C n peut s’exprimer en fonction d’une somme exponentielle. Nous montrerons que cette somme n’atteint pas la borne de Weil et nous proposerons une amélioration de celle-ci. En conséquence, nous obtiendrons une amélioration de la borne de Carlitz-Uchiyama sur le poids des mots du dual de C n .

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