**PD-sets for partial permutation decoding of Z**_{2s} -linear Hadamard codes

_{2s}-linear Hadamard codes

### Merce Villanueva,

**Autonomous University of Barcelona **

Date: Thursday 5th September

Time: 1:00pm–2:00pm (Talk & Q/A)

Venue: Building 8 Level 9 Room 66 (AGR) RMIT City campus

The seminar will be followed by snacks and drinks

All students, staff and visitors are welcome

**ABSTRACT:**

Let *Z _{2s}* be the ring of integers modulo

*2*with

^{s}*s≥1*, and let

*Z*be the set of

_{2s}^{n}n-tuples over

*Z*. A nonempty subset

_{2s}*C*of

*Z*is a

_{2s}^{n}*Z*-additive code if

_{2s}*C*is a subgroup of

*Z*. Note that, when

_{2s}^{n}*s=1*,

*C*is a binary linear code; and when

*s=2*, it is a quaternary linear code or a linear code over

*Z*.

_{4}The

*Z*-additive codes can be seen as binary codes (not necessarily linear) under a generalization of the usual Gray map,

_{2s}*Φ:Z*. The binary image

_{2s}^{n}⇒Z_{2}^{n2s-1}*C=Φ(C) is a Z*-linear code of length

_{2s}*n2*. Permutation decoding is a technique, first introduced for linear codes, that involves finding a special subset, called a PD-set, of the automorphism group of a code. In 2015, a new permutation decoding method for

^{s-1}*Z*-linear codes and, in general, for systematic codes (not necessarily linear) was introduced, but the determination of PD-sets for nonlinear codes remained an open problem. In 2018,

_{4}*s*-PD-sets of minimum size

*s+1*for some families of nonlinear systematic codes such as

*Z*-linear Hadamard, Kerdock and simplex codes were given. We will show the generalization of some of these results to the family of

_{4}*Z*-linear Hadamard codes.

_{2s}**BIOGRAPHY:** Mercè Villanueva was born in Roses, Catalonia, in January 1972. She received the B.Sc. degree in Mathematics in 1994 from the Autonomous University of Barcelona, the M.Sc. degree in Computer Science in 1996, and the Ph.D. degree in Science (Computer Science Section) in 2001 from the same university. In 1994 she joined the Department of Information and Communications Engineering, at the Autonomous University of Barcelona, as an Assistant Professor, and was promoted to Associate Professor in 2002. Her research interests include subjects related to combinatorics, algebra, coding theory, and graph theory.