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Bounds on the rank and kernel of perfect codes
BZtIR Bt Y st, stI ,tj BP }P{ {BIR %$t q _Yj}R
VZAZt =t$%R$ k {r z$jjstZ%s
= VZ rBtB8s I s{jBts V A$tsk {BI g BP jt Y z $R `ZR s R BP A$tsk zT Z}jR g 2 B z B ) \(, di V j$ts wA$tskD {BI $R `ZR s j$ts RZAR}s{ BP zTI$8tR$Btsj A$tsk R}s{ V }P{ R$tj B {B{ $t A$tsk {BI wA$4k dT}P{ {BID $R s RZAR BP B z RZ{Y Ys %k ? ; B z b$ Y$t I$R st{ d PB8 -s{ jk Bt {BIbBI BP g y YsR jt Y z ) 1E d 8$t$8Z8 I$R st{ n stI YsR 1z E {BIbBIR N$ts dT}P{ {BIR BwgD B Y Y st, stI ,tj BP dT}P{ {BIR Ys% At R ZI$I e 9$Bt stI zsIk >dH R sAj$RYI Y -$R t{ BP s dT}P{ {BI BP jt Y z ) 1E d z dx stI st, wgD ) z E L X PB s{Y X X ) (, d, 3 3 3 , E _Yj}R stI Nzst >;H R sAj$RYI Ys PB s{Y RZ{Y z dx Y -$R R s tBtj$ts dT }P{ {BI g BP jt Y z b$ Y s ,tj BP I$8tR$Bt >wgD ) > PB s{Y > ; \d, 1, 3 3 3 , z E 1i qY st, stI ,tj s ,tBbt B A js I 3B $tR st{ b Ys% >wgD L wgD z L d G R sAj$RY Z}} stI jBb ABZtIR Bt Y st, stI Y I$8tR$Bt BP Y ,tj BP dT}P{ A$tsk {BIR
_}$t RZA8$ I B ejR%$ 7{$t{
235
qY jBb ABZtI Bt Y I$8tR$Bt BP Y ,tj $t 8R BP Y st, BP Y {BI $R $%t Ak Y PBjjBb$t RZj R stI $ $R -s{ qYB8 d V dT}P{ {BI g BP jt Y z ) 1E d stI Ys%$t st, wgD ) z E L X stI s ,tj BP I$8tR$Bt >wgD Yt >wgD
1E X
7W X k d
>wgD 1E @ d 7W X ) d
qYB8 1 >;H 3B sjj E ; Y -$R R s dT}P{ {BI BP jt Y z ) 1E d Ys%$t st, z E L X stI ,tj BP I$8tR$Bt > ) 1E X bYt X k d stI > ) 1E @ d bYt X ) d qY Z}} ABZtI Bt Y I$8tR$Bt BP Y ,tj BP s dT}P{ {BI $t 8R BP Y st, BP Y {BI $R $%t Ak Y PBjjBb$t RZj qYB8 n V dT}P{ {BI BP jt Y z ) 1E d Ys YsR st, z E L X stI s ,tj BP I$8tR$Bt z E & Yt 1& & d X e 9$Bt stI zsIk >1H $% s {BtR Z{ $Bt BP PZjj st, dT}P{ {BIR w$ X ) ED Ys s{Y$% Y$R ABZtI bYt E d( qY RZj R PB8 _Yj}R >nH RYBb Ys Y$R ABZtI $R $Y PB E ) ; stI X c ; qB R sAj$RY Y$R Z}} ABZtI b b$jj tI RB8 RZj R Bt
x k
7; {
7
yzQ Q7EwI@ I D ) E L X
bY { $R s R BP E & $tI}tIt {BBI$ts R $P 1& & d X stI & c E qYB8 x p$%t
x k
7; {
7
yzQ Q7EwI@ I D ) E L X
Y -$R R s dT}P{ {BI g BP jt Y zI ) 1E d bY EI ) E L d Ys YsR st, zI EI L X stI s ,tj BP I$8tR$Bt zI EI & $P & $R Y 8$t$8Z8 $t RZ{Y Ys 1& & d X 1
236
WB ${ Ys $P E ; X ; \d, 3 3 3 , Ei Y -$R R s jsR Bt & RZ{Y Ys 1 & d X stI & c E 7B N88s ; stI qYB8 x $% ZR Ys Y Z}} ABZtI $R $Y E x stI X c E qB }B% Y }%$BZR qYB8 b tI B R sAj$RY RB8 RZj R Bt Y st, stI Y ,tj BP dT}P{ {BIR {BtR Z{ I b$ Y Y *BZAj$t {BtR Z{ $Bt
iPt{R
>dH
q
e 9$Bt
V
zsIk
yeee qstR Bt ytPB8s $Bt qYBk
>1H
q e 9$Bt V zsIk
7yV F *$R{ s Y >nH
q
_Yj}R
;(wd;D x;TEn
ddwdMD tB 1 1(xT11n
! "
1dw1(((D 1MT1M
VZR sjsR$st FBZtsj BP ]B8A$ts B${R >;H
q _Yj}R Nzst stI ]k} Bs}Yk
# $
*R$tR ]BIR
EwdxD 1;T1x
237
VZAZt =t$%R$ k {r z$jjstZ%s
= VZ rBtB8s I s{jBts V A$tsk {BI g BP jt Y z $R `ZR s R BP A$tsk zT Z}jR g 2 B z B ) \(, di V j$ts wA$tskD {BI $R `ZR s j$ts RZAR}s{ BP zTI$8tR$Btsj A$tsk R}s{ V }P{ R$tj B {B{ $t A$tsk {BI wA$4k dT}P{ {BID $R s RZAR BP B z RZ{Y Ys %k ? ; B z b$ Y$t I$R st{ d PB8 -s{ jk Bt {BIbBI BP g y YsR jt Y z ) 1E d 8$t$8Z8 I$R st{ n stI YsR 1z E {BIbBIR N$ts dT}P{ {BIR B
_}$t RZA8$ I B ejR%$ 7{$t{
235
qY jBb ABZtI Bt Y I$8tR$Bt BP Y ,tj $t 8R BP Y st, BP Y {BI $R $%t Ak Y PBjjBb$t RZj R stI $ $R -s{ qYB8 d V dT}P{ {BI g BP jt Y z ) 1E d stI Ys%$t st, wgD ) z E L X stI s ,tj BP I$8tR$Bt >wgD Yt >wgD
1E X
7W X k d
>wgD 1E @ d 7W X ) d
qYB8 1 >;H 3B sjj E ; Y -$R R s dT}P{ {BI BP jt Y z ) 1E d Ys%$t st, z E L X stI ,tj BP I$8tR$Bt > ) 1E X bYt X k d stI > ) 1E @ d bYt X ) d qY Z}} ABZtI Bt Y I$8tR$Bt BP Y ,tj BP s dT}P{ {BI $t 8R BP Y st, BP Y {BI $R $%t Ak Y PBjjBb$t RZj qYB8 n V dT}P{ {BI BP jt Y z ) 1E d Ys YsR st, z E L X stI s ,tj BP I$8tR$Bt z E & Yt 1& & d X e 9$Bt stI zsIk >1H $% s {BtR Z{ $Bt BP PZjj st, dT}P{ {BIR w$ X ) ED Ys s{Y$% Y$R ABZtI bYt E d( qY RZj R PB8 _Yj}R >nH RYBb Ys Y$R ABZtI $R $Y PB E ) ; stI X c ; qB R sAj$RY Y$R Z}} ABZtI b b$jj tI RB8 RZj R Bt
x k
7; {
7
yzQ Q7EwI@ I D ) E L X
bY { $R s R BP E & $tI}tIt {BBI$ts R $P 1& & d X stI & c E qYB8 x p$%t
x k
7; {
7
yzQ Q7EwI@ I D ) E L X
Y -$R R s dT}P{ {BI g BP jt Y zI ) 1E d bY EI ) E L d Ys YsR st, zI EI L X stI s ,tj BP I$8tR$Bt zI EI & $P & $R Y 8$t$8Z8 $t RZ{Y Ys 1& & d X 1
236
WB ${ Ys $P E ; X ; \d, 3 3 3 , Ei Y -$R R s jsR Bt & RZ{Y Ys 1 & d X stI & c E 7B N88s ; stI qYB8 x $% ZR Ys Y Z}} ABZtI $R $Y E x stI X c E qB }B% Y }%$BZR qYB8 b tI B R sAj$RY RB8 RZj R Bt Y st, stI Y ,tj BP dT}P{ {BIR {BtR Z{ I b$ Y Y *BZAj$t {BtR Z{ $Bt
iPt{R
>dH
q
e 9$Bt
V
zsIk
yeee qstR Bt ytPB8s $Bt qYBk
>1H
q e 9$Bt V zsIk
7yV F *$R{ s Y >nH
q
_Yj}R
;(wd;D x;TEn
ddwdMD tB 1 1(xT11n
! "
1dw1(((D 1MT1M
VZR sjsR$st FBZtsj BP ]B8A$ts B${R >;H
q _Yj}R Nzst stI ]k} Bs}Yk
# $
*R$tR ]BIR
EwdxD 1;T1x
237