Codes of the triangular graphs

Permutation Decoding

10.2 Codes of the triangular graphs

In this section we describe specific PD-sets for the binary codes obtained from the triangular graphs. For decoding purposes we order the points in such a way that the nature of the information symbols is known and the action of the automorphism group apparent. These codes were constructed in Chapter 8 and their properties were given in Lemma 8.2.3. In Lemma 8.2.7 we showed that by ordering the points in the following way:

{I, 2}, {I, 3}, ... , {I,

n -

I}, {2, 3}, ... , {2,

n -

I}, ... ,

{n -

2,

n -

I}, (10.1) followed by the remaining points

{I, n}, {2, n}, ... , {n - 1, n}

we get the generator matrix of Cl.. in upper triangular form.

(10.2)

The generator matrix obtained in Lemma 8.2.7 for Cl.. with the above ordering can be reduced to the form [hIA) where k is the dimension of Cl... Ifthe points are re-ordered with the first k put at the end, then the matrix is [Alh). This is now standard form for the code C, and the corresponding generator matrix for C has the form

[I

^{n -} k

I

AT], where here we are using n for the length of the code.

In order to get the generator matrix into standard form, as described above, we order the point set P by taking the set from Equation (10.2), that is

PI = {I, n},P2 = {2, n}, ... ,P_{n - 1} =

{n - 1, n},

_(10.3)

first, followed by the set from Equation (10.1), that is

Pn

=

{1, 2},Pn+1

=

{1,3}, ... ,P2n- 2

=

{2,3}, ...,p(~)

= {n - 2, n - 1}.

(10.4) The generator matrix for Cl., using the words of weight 3 (with J if n is even), is then a check matrix for C in standard form. The generator matrix for C will then also be in standard form, with the first n - 1 coordinates the information symbols for n odd, and the first n - 2 for n even. Using the fact that the group of the code is Sn, we can find PD-sets for the code C.

Proposition 10.2.1 For n 2:: 5 odd, a PD-set of n elements can be found for C.

If the points are ordered as given in Equations (10.3) and (10.4), the set S={le}U{(i,n) 11:Si:sn-1}

of permutations in Sn in the natural action on the points P forms a PD-set of n elements for

c.

Proof: Order the points of the coordinate set P as described in Equations (10.3) and (10.4) so that the first n - 1 points are in the information positions.

Now C can correct

t =

n~3 errors. We need a set S of elements of G

=

Sn

=

Aut(C) such that every t-set of elements of P is moved by some element ofS into the check positions. Ifthe s:S

t

positions are all in the check positions, then we can use the identity element, le, to keep these in the check positions.

Suppose the s :S

t

positions occur at

distinct points in the information positions, and at

distinct points in the check positions, where

r +

m

= s < t.

The number of elements of

n

in the set

T =

{aI, ... , a_{r }} U

{b

l , ... ,

b

m } U{Cl, ... ,Cm} <;;;

n \

{n}

is at most r

+

2m. Since r

+

m :::;

t =

(n - 3)/2, we have 2r

+

2m :::; n - 3, and thus r

+

2m :::; n - 3. Thus there are elements other than n in

n

that are not in

T;

let

d

be one of these. The transposition ^(j

= (d, n)

will map the

r

elements

out of the information positions, as required, and fix the m elements already in the check positions.

It follows that the given set

S

=

{le} U {(i,n) 11:::;

i:::;

n -1}

forms a PD-set ofn group elements for the code. •

(10.5)

In this case the Gordon bound (see Theorem 3.5.5) has an explicit form:

Lemma 10.2.2 For n _~ 5 odd, the Cordon bound for C is _n~l.

Proof: The length of the code is

G)

and the redundancy is r

= G) -

ⁿ

+

1. With

t =

n~3, we have

G) -

^t

+

1 n² - 2n

+

5

(~)

^{- n}

+

1 -

t +

1

=

n 2 - 4n

+

7 for the innermost term. In fact the Gordon bound is

r

_{n 2 - 3n}^{n2 - n}

₊

_{2 ...}

r r

ⁿ²_{n2 -}^- _4n²ⁿ

⁺ ₊

⁷₉

r

_nⁿ²_{2 -}^{- 2n}_4n

⁺ ₊ 511 11

_{7 . . . .}

It is not hard to show that this is equal to n~l for n ~ 5 and odd. •

Proposition 10.2.3 For n 2: 6 and even, a PD-set of n^{2 -} 2n

+

2 elements can be found for C. If the points are ordered as given in Equations (10.3) and (10.4), the set

5

=

^{{le} U {(i,n)} 11 :S i:S n -1}^U {[(i,n -l)(j,n)]±l11:S i,j:S n - 2}

of permutations in Sn in the natural action on the points P is a PD-set for C.

Proof: Again we order the points as in Equations (10.3) and (10.4) so that now the points PI, P_{2 , . .. ,}P_{n -}2 are in the information positions,

I,

and the remaining points of

P,

starting with

P

n - I

=

{n -1,n}, then followed by

P

n , . . . ,p(~), are in the check positions, £. In this case we need to correct t

=

^{n -} 3 errors, since the minimum weight is 2(n - 2).

We claim that

5

=

{le} U {(i,n) 11:S i:S n -1} ^U{[(i,n -l)(j,n)]±1 11 :S i,j:S n - 2}

is a PD-set for C. Note that 151

=

1+n-1+2(n-2)+(n-2)(n-3)

=

n²-2n+2.

We need to show that everyt-tuple

T

of points of

P

can be moved into the check positions £ by some member of 5. Consider the various cases for the members of T:

(i): ifall the t positions are in£ then le will do;

(ii): if all the t positions are inI then (n - 1,n) will do;

(iii): if some a E

n \

{n} does not occur in any member ofT then (a, n) will do.

We can thus restrict our attention to those sets T for which every a E

n

appears in some duad in T. We show that if

{a,

b} E T and

a

does not occur again in any element of T, then an element of 5 can be found to map T into £.

Consider the possible cases:

(iv): a

=

nand b

=

n - 1, then le will do; if b

=J

n - 1, then (b, n - 1) will do;

(v): a

=J

nand b= n then if a = n - 1, (n, n - 1) will do and if a

=J

n - 1 then

(a, n, n - 1)

=

(a, n)(a, n - 1) will do;

(vi): a

=I

nand b

=I

n then if a

=

n - 1, (b, n - l)(b, n) will do; if a

=I

n - 1, then if b

=

n - 1, (a, n) will do and if b

=I

n - 1, (a, n)(b, n - 1) will do.

So if there is a duad

{a, b}

E T such that

a

occurs only once, our set of permutations will form a PD-set. Now every a E

n

occurs and if every element appears more than once we would have 2n elements to place in 2t

=

2(n - 3) positions, which is impossible. •

Remark 10.2.4 (i) The code Cl.. has minimum weight 3, so can only be used for single-error correction. Thus syndrome decoding would be the usual method employed. However notice that PD-sets can be found easily for this code too, using the ordering of the points given in Lemma 8.2.7, where the set S of n permutations given in Equation (10.5) will form a PD-set for Cl.. for n ~ 5 odd or even. The Gordon bound is less than this number.

(ii) The permutations given in the set S need to be written as permutations on the points

g,

P2 ... ,p(~). Thus, for example, ifn

=

6, then with the ordering of the points as given in Equations (10.3) and (10.4),

(1,6) (1,5)(1,6) (1,5)(2,6)

(P2,P6)(P3,P7)(P4,PS)(P9,P5)

(PI,P9,P5)(P2,P6,PI2)(P3,P7,PI4)(P4,PS,PI5) (PI, PI2 )(P3,PlO) (P4,P

n

)(P6,P5)(P7,PI4 )(PS,P15 )

(iii) For n even the Gordon bound becomes

r

_{n 2 - 3n}^{n2 - n}

₊

₄

r

_{n 2 - 3n}^{n2- n -}

₊

²_{2 ...}

r r

_{n 2 - 5n}^{n2- 3n}

₊ ⁺

₁₂

811 11

_{. . . .}

An exact formula for this, in contrast to the odd case, does not seem evident, but from computations (using Magma [11]) up to a large value ofn, the following formula appears to hold for this bound for n ~ 18 (smaller values of n seem

to be unrepresentative of the general rule): writing n = 2k

+

18, k1 k (mod 6) E {O,1,2,3,4,5}, k ~ 0, the Gordon bound for n is

n

+

8

+

10

l ~ J ⁺

^k1

⁺ l ~l J.

For n even the size of the PD-sets we have found are of the order of n², a lot bigger than the Gordon bound, which gives the order ofn; the Magma output in Appendix E, illustrates this. However, we show also the size of the automorphism group in comparison to illustrate that our sets are a lot better than trying to use the whole group.