Weight distribution of a Reed-Muller code

We illustrate some of the elementary coding theory facilities by constructing the second-order binary Reed-Muller code of length 128. We then determine its number of code words, minimum weight, complete weight distribution and weight enumerator. Finally, we construct the dual code and compute the same information.

> R := ReedMullerCode(2, 7);
> #R;
536870912
> MinimumWeight(R);
32
> WeightDistribution(R);
[ <0, 1>, <32, 10668>, <48, 5291328>, <56, 112881664>, <64, 300503590>, 
<72, 112881664>, <80, 5291328>, <96, 10668>, <128, 1> ]
>
> f<x, y> := WeightEnumerator(R);
> f;
x^128 + 10668*x^96*y^32 + 5291328*x^80*y^48 + 112881664*x^72*y^56 + 
    300503590*x^64*y^64 + 112881664*x^56*y^72 + 5291328*x^48*y^80 + 
    10668*x^32*y^96 + y^128
> D := Dual(R);
> #D;
633825300114114700748351602688
> WeightDistribution(D);
[ <0, 1>, <8, 188976>, 
<12, 148157184>, <14, 5805342720>, 
<16, 352501184760>, <18, 14090340827136>, 
<20, 445990551166720>, <22, 11148730324353024>, 
<24, 224814298345622160>, <26, 3704888469231108096>, 
<28, 50486579825291883008>, <30, 574502111223143792640>, 
<32, 5505259862572668584988>, <34, 44748635843913605775360>, 
<36, 310470295870406870385152>, <38, 1848689416882328323358720>, 
<40, 9492309127074743252712240>, <42, 42202740208778987487756288>, 
<44, 163056041735354833829648640>, <46, 549191653630903808742490112>, 
<48, 1616902022777436781296463560>, <50, 4170947258549850556429074432>, 
<52, 9445968792148616532912076032>, <54, 18812726104570634921033072640>, 
<56, 32995567020448757300816680976>, <58, 51020368602507380313683656704>, 
<60, 69612536825673328395392461824>, <62, 83858994648178551820509904896>, 
<64, 89224971989924438343276144710>, <66, 83858994648178551820509904896>, 
<68, 69612536825673328395392461824>, <70, 51020368602507380313683656704>, 
<72, 32995567020448757300816680976>, <74, 18812726104570634921033072640>, 
<76, 9445968792148616532912076032>, <78, 4170947258549850556429074432>, 
<80, 1616902022777436781296463560>, <82, 549191653630903808742490112>, 
<84, 163056041735354833829648640>, <86, 42202740208778987487756288>, 
<88, 9492309127074743252712240>, <90, 1848689416882328323358720>, 
<92, 310470295870406870385152>, <94, 44748635843913605775360>, 
<96, 5505259862572668584988>, <98, 574502111223143792640>, 
<100, 50486579825291883008>, <102, 3704888469231108096>, 
<104, 224814298345622160>, <106, 11148730324353024>, 
<108, 445990551166720>, <110, 14090340827136>, 
<112, 352501184760>, <114, 5805342720>, 
<116, 148157184>, <120, 188976>, 
<128, 1> ]

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