Some non-existence theorems for perfect codes over arbitrary alphabets

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Some non-existence theorems for perfect codes over arbitrary
alphabets
Citation for published version (APA):
Reuvers, H. F. H. (1977). Some non-existence theorems for perfect codes over arbitrary alphabets. [Phd Thesis
1 (Research TU/e / Graduation TU/e), Mathematics and Computer Science]. Technische Hogeschool
Eindhoven. https://doi.org/10.6100/IR140570
DOI:
10.6100/IR140570
Document status and date:
Published: 01/01/1977
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SOME NON -EXISTENCE THEOREMS
FOR PERFECT CODES
OVER ARBITRARY ALPHABETS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE
TECHNISCHE WET ENS CHAPP EN AAN DE TECHNISCHE
HOGESCHOoL EINDHOVEN, oP GEZAG VAN DE RECTOR
MAGNIFICUS, PROF.Dr. P. VAN DER LEEDEN. VOOR
EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN
DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
OINSDAG 18 JANUARI 1977 TE 16.00 UUR
door
HENRICUS FRANCISCUS HUBERTUS REUVERS
GEBOREN TE MAASTRICHT
Pit ~oe6J~~rit ~ goedge~e~d
doo~ de ~omoto4en
P40n. dt. J.H. van Lint
P40n. ~. J.H. de Bo~
DRUKKERIJ UIHl.V.AIJ BIJSTERVELD
EINDHOVEN
'IDOft M.vUtm en de Uurte Wk
/lNI. m~jr. "t.«/vu
CON'I'llNTS
C!lJU>'I'llR 1. ImROJ:)UCTION
1.1. On error-cQ~recting eQde~
1.2. on perfect codes and the sphere packing condition 2
1.3. On the n~er of code words of weight k for some small k
1.4. On t-deaigns 4
1.5. On t-de~igns in perfeot codes 5
1.6. The polynomial condition 6
1.7. Some example~ of perfect code~ 11
1.e. A remar~ about perfeot 1-codes 12
1.9. S~ry of results 15
CHlU>'I'ER 2. SOME GENE:AA.l.. RESULTS CONCEI\NWG q 17
s
2.1- The general cast!! q; " P (e <: 2) 17
s t
2.2. The general Case q '" P1P2 (e ;!; 3) 19
2.3. Introauction to a result concerning the number of primes dividing q 20
2.4. Statement of a result concerning the number of primes dividing q 23
:2.5. Application to the case e = 6 26
~llR 3. SOME RESULTS CONCERNING THE CAsE e 27
3.1. First approach to the case e = 2 27
3.2. Second approaoh to the case e '" 2 30
3.3. The special Oase q '" pS 32
s t
3.4. A remark abol,1t the special caee Q. = P1P2 2\5 33
3.5. Some remarks about the special .ease q = 34
3.6, The special case q < 30 or q '" 3S 37
3.7. The special case q .. 30 53
CHAPTER 4. S~ GENERAL RESULTS CONCERNING n 71
4.1. A first remark about n 71
4.2. A second remark about n 72
page
4.3. ~he e~istence of an upper ~o~nQ N(e,q) for n 74
4.4. ~ u~per bound N(e.q) maae explicit if @ is odQ 76
4.5. Application 81
CHAP'rIlR 5. SOMil .RESULTS FOR SMALL VALTJ:o:S OF e B2
5.1. 'rhe case e 3 82
5.2. The case e = 4 83
5.3. The case e = 5 88
CHAPTER 6. SOME R5SULTS CONCERNING PERF~cr MIX£D CODES 91
6.1. preliminar~e5 91
6.2. A none~istence the~~ concerning mixed pe~fect 2-codes 93
6.3. A nonexi~tenoe theorem concerning mixed perfect 3-codes 99
APPENDIX 102
HISTORICAL SUMMARY 116
REFERENCES 120
SAMENVATTING 125
CURRICULUM VI'r~ 127
CHAPTER 1 : TNTROVUCTION
Let S:be a set of q symbols. we shall take s := {O, 1 ,2, ... ,q - l}.
We call S an alphabet.
Let, for same n € IN, V be the Cartesian product Sn. We call V a spacl'l,
and the elements of V wo:r>as.
Let C be a subset of V. Then we call C a codl'l. The elements of C are
called code wo:r>d8 and n is called the wo:r>d ~ength of C.
C is called a ~oup code if it is a group Under coordinatewise addition
(moOulo q).
Let x € V. Th@n the Hamming weight Wn(~) of ~ is the number of cocrdinate
places in which ~ has a nonzero sywbol.
The ~«ppOrt of ~ is the vector supp (~) which has zeros in e~actly the
same coordinate places in which ~ has ~er08, and which has ones in the
other coordinate pLaces.
FOr any two words x and l in the space V, we define thQ Hamming distance
dH(~'~) between x and Z to be the number of ccordinate places in which ~
and ~ have a different symbol, so
1.1.1.
where subtraction means ccordinatewise subtraction (mea q) •
By dH(~'C) we denote the distanoe from x to the oode.
FOr any x ~ V, let the Hamming sphere W~th radius e arOund x be defined :by'
1.1.2.
A cOde C is caned (?-error-coX'recting (;fOr some e € IN) if for any two
distinct code words ~ and :i we have dH(~.r) ~ 2e + 1, 50 if the spheres
With radius e around the oode WOrds are di5joint.
If C i6 an e-error-correctlng cod~, ~~d if W~ ahange at mo~t e ooordinates
2
of a cooe word ~, then the changed word is still nearer to ~ than to any
other code woro.
~ cooe C is called e-error-detecting if, for any two 01stinot code woros
~ and ~, we have dH(~,r) ~ 2e.
The numbers, n, e ana q are called the traditional parametep8 of a code.
We call a code c a perfect e-code if the Hamming spheres with radJ.\18 e
around the code words form a partition Of V.
Such a code is not only e-error-correcting, but the Hamming spheres fill
the space V.
rt wae proved by Lenstra (see [19]) that a perfect code over Ii q-symbol
~lphabet cannot be a group code unless q is a power of a prime.
The case where q is a power of a prime was settled completely oy Van
Lint and ~ietava1nen (see seotions 1.7, 2.1 and the historical survey).
They proveo that unknown perfect codes over GF{q) do not exist.
~xampleB of perfeCt codes can be found in section 1.7.
It is O\1r pu~po?e to prove nonexistence theorems for perfect codes wLth
p~r~metcrs n, e, q, where q is not necessarily a power of a prime. In this
case we call S dn arbitrary alphabet.
~ oovious necessary condition for tne existence of perfect codes is
called the sphere paakil1lJ aondition:
Here the left hand side is the n~mber of words in a sphere with radius e,
and qn is the total number of words in the space V.
If, to~ ~nstance, q ~s ~ prime power, sav q = po, then we have the ve~y
str-ong condition that for SOme a " IN
3
1. 2.2.
From now on th~ symbol ~ will denote ~ p~ime.
The sphere packing condition ~lays a basic role in our investi~ations.
Ass~e that we have a perfect code with parameters n, e, q. we can
assume without loss of gener~lity that the word Q (h~ving a zero in
eve~y cocrdina te place) is a cede word.
In this case the minimum weight of a cod~ word is 2~ + 1.
~hen each word of weight e + 1 is in exactly one Hamming sphere with
radius e around a code word, and this code word must pe of wei~ht
2e + 1.
n e+l
Therefore, since the~e are (e+l) (q - 1) words of weight e + 1, and
since in a Hamming 8~here with r~d!us e around a code word of wei~ht
2a+l
2e + 1 there are (e+l) words of weight ~ + 1, we find that the number
a + of code words of weight 2e + 1 must be:
2e 1
1.3.1.
~urthermore, since each word of weight e + 2 is in exactly one Hamming
~~here with radius e a~ound ~ code wo~d, and th!s c04e wo~d must be of
weight 2e + or 2~ + 2, we find, counting the words of weight e + 2
~t ~ distance e o~ e - 1 from such a code word, the following ~ecur~ence
relation, which determines the number a + of code words of weight 2e + 2:
2e 2
2e+2 210+1 2e+1
1.3.2. ( e+2 ) a2e+2 + (10+2) a2e+1 + (e+l) e(q - 2)a2e+1
which yields'
2
(n - e - 3e - 1) (9 - 1) + e (e + 1)
1. 3. 3. 2e + 2 a2e+1
4
In the same Wi!J.y we car. Q.ete=ine the nwnber "'2e+3 of code words of
weight 2e + ~ Py means of the following reCUrrenCe re~",tion:
1. 3.4.
In this way we can go on.
So the numbers a depend on k, n, e and q.
k
For our purpose we shall onll'" need the",e nwnbers for the case e .. 4,
In the appendix (see A.l) we determine the numbers a for e = 4 and
1
9 :;; i :;; 13.
1.4. On -t-de1>igYL6
For the sections 1.4 dnd 1.S We refer to (41J, chapter 2, section 4.
We say that a word Z ~ V coverB another word x if we hav",
so if under coordinatewise multiplication we have ~'5Upp(~) ~ x.
NOW we define a (q-ary) design of type t-(n, k, A) in V to be a
collection D of ·words of weight k in V, such that every word ot
weight t in v is covered bl'" e~ctly A members of D.
This definition generalizes the concept of binary t~de5i9nB (see [27J)
to the concept of q-ary t-designs.
If in V there exists a q-ary design of type t-(n, k, ~), then tr!vially
~ must·be an integer. ~ut moreover,
1.4.2. ~or 0 ~ i :;; t we have Ai ~ ~, where
1) t-i
. \ :=