Hadamard matrices from the Multiplication Table of the Finite Fields

Transcription

1 adamard marice from he Muliplicaio Table of he Fiie Field 신민호 송홍엽 노종선 *

2 Iroducio adamard mari biary m-equece New Corucio Coe Theorem. Corucio wih caoical bai Theorem. Corucio wih ay bai Remark

3 adamard mari Iroducio Defiiio : A adamard mari of order i a by mari wih erie or - uch ha T I Eample. adamard mari of order Noe Ay wo row of are orhogoal. hi propery doe o chage if we permue row or colum or if we muliply ome row or colum by - Noe Two uch adamard marice are called equivale. deoe - deoe -

4 Relaio bewee adamard marice ad ECC All he row of a adamard mari of order form a orhogoal code of legh ad ize. All he row of a adamard mari of order ad heir compleme form a biorhogoal code of legh ad ize All he row of a ormalized adamard mari of order wihou heir fir compoe form a imple code of legh - ad ize

5 m-equece Defiiio : Maimal legh LFSRLiear Feedback Shif Regier equece A Liear recurrig equece degree m over Fq wih recurrece relaio m i a i i F ca be geeraed by a m-age LFSR wih a characeriic polyomial a i q f m a m a m a m Eample. Geeraio of a biary m-equece wih -age LFSR liear recurrece degree characeriic polyomial ha a period f 7

6 m-equececo d Fac A LFSR produce a m-equece over GFq if ad oly if i characeriic polyomial i primiive i GFq m-equece are aalyically repreeed by he race fucio Properieeleced r GF q {} where race fucio r map GF q io GFq auocorrelaio properybiary equece of period N : primiive i GF q N N mod N b mod N cycle ad add propery : he um of m-equece { } ad i -hif { } i aoher hif { } of he ame m-equece

7 Relaio bewee adamard marice ad biary m-equece Eample. m-equece period v adamard mariorder { } } } { { Cyclic adamard mari order 8 " " "" r 7

8 Relaio i geeral } biary m-equece of { period N Wih race repreeaio N N r GF : primiive i {} GF N by N mari - maric : N by N circula mari geeraed by cyclic hif of } Wih race repreeaio C c i j N c ij { ij r C i j By auocorrelaio propery of he m-equece do produc of ay wo row of N by N mari C i -afer chagig o o - ece N by N mari defied a above i a adamard mari of order 8

9 9 New corucio Corucio i GF Eample. From muliplicaio able of wih caoical bai. : primiive i aifyig GF GF Field geeraio Muliplicaio able Noe each ucceive equece vecor repreeed from h coefficie i cyclically equivale m-equece. Noe by mari i circula. i Polyomial Vecor Power

10 Eample. co d adamard marice from he vecor repreeed muliplicaio able of caoical bai r r r

11 Theorem. Le GF be he fiie field wih eleme ad GF be a primiive eleme. Coider he muliplicaio able of GF wih border. Le he erie or hi able be vecor-repreeed over caoical bai GF uig he. For i le be he mari obaied by akig he i-h compoe of all he erie of he muliplicaio able. The hee marice i oly by colum permuaio i are adamard marice ad hey are equivale

12 Eample. From muliplicaio able of wih arbirary bai. : primiive i aifyig chage coordiae from caoical bai o he bai by caoical bai epaio ad bai epaio Defie biary row vecor ad by Le bai arbirary From above relaio defie by marice A ad B The we ca chage he coordiae a follow GF GF GF B A B B A

13 Power Check Eample. co d r r r r r r r r r r r r r Caoical bai bai

14 Eample. co d Caoical bai bai r r U U T U Noe he raformaio mari U i a permuaio mari. i.e ece wo uch marice are equivale by rowor colum permuaio UU T I

15 Theorem. Repreeaio of eleme i ay bai. GF i Theorem ca be doe by uig Relaio of adamard marice ad m-equece caoical bai The i r i bai ca be repreeed by Defie he by mari B b ij ad A B a ij The are relaed o he m-equece a follow bai i j i b ij i b ij { } j i k a ki r k r aki k r k i Noe k i a ki k GF {}

16 Remark. No-bai repreeaio may o work Eample 7. A mari obaied from he o-bai vecor repreeed muliplicaio able of Noe h row ad h row are o orhogoal GF Remark row h row h

17 Remark. The followig cojecure i fale Coider arbirary umber of adamard marice If i i a adamard mari where mari addiio i compoewie mod The i are m-equece adamard marice Couer eample. Coider GMW equeceg r r 7

18 Remark. are liearly idepede over GF. Sice { r } i oe of LFSR oe ca fid i { for ome } { GF {} ece hey are liearly idepede. } 8