Resumen de On additive binary nonlinear codes and steganography

Lorena Ronquillo Moreno

• A code C is said to be Z2Z4-additive if its coordinates can be partitioned into two subsets X and Y , in such a way that the punctured code of C obtained by removing the coordinates outside X - or, respectively, Y - is a binary linear code - respectively, a quaternary linear code -. The Gray map image of C is a binary and often nonlinear code called Z2Z4-linear code. In this dissertation, new families of Z2Z4-additive codes are presented, with the particularity that their Gray map images are Z2Z4-linear codes having the same parameters and properties as the well-known family of binary linear Reed-Muller codes. Considering the class of perfect Z2Z4-linear codes, which are known to be completely regular, we have used the extension, puncture, shorten and lifting constructions, and studied the uniformly packed condition and completely regularity of the obtained codes. Besides providing reliability in communication channels, coding theory has been recently applied to steganography, i.e., the science of hiding sensitive information within an innocuous-looking message -the cover object- in such a way that third parties cannot detect that information. This hiding process has been addressed in the literature either by distorting the least significant bit of symbols in the cover object to transmit the secret message (binary steganography), or by distorting the two least significant bits (+/-1-steganography). With respect to +/-1-steganography, two new embedding methods based on perfect Z2Z4-linear codes are introduced, achieving a higher embedding rate for a given distortion than previous methods; while another method, based on the product of more than two perfect q-ary Hamming codes, is presented conforming to binary steganography.