Weight Distributions

In the case of a linear code, weight and distance distributions are equivalent (in particular minimum weight and minimum distance are equivalent).

Hamming Weight

For an element \(x\in R\) for any finite ring \(R\), the Hamming weight \(w_H(x)\) is defined by:

\[w_H(x) = 0 \iff x = 0, \qquad w_H(x) = 1 \iff x \ne 0\]

The Hamming weight \(w_H(v)\) of a vector \(v\in {R^n}\) is defined to be the sum (in \({\mathbb{Z}}\)) of the Hamming weights of its components.

The Hamming weight is often referred to as simply the weight.

MinimumWeight(C): Code -> RngIntElt
MinimumDistance(C): Code -> RngIntElt: Determine the minimum (Hamming) weight of the words belonging to the code \(C\), which is also the minimum distance between any two codewords.

WeightDistribution(C): Code -> [ <RngIntElt, RngIntElt> ]: Determine the (Hamming) weight distribution for the code \(C\). The distribution is returned in the form of a sequence of tuples, where the \(i\)-th tuple contains the \(i\)-th weight, \(w_i\) say, and the number of codewords having weight \(w_i\).

DualWeightDistribution(C): Code -> [ <RngIntElt, RngIntElt> ]: Determine the (Hamming) weight distribution of the dual code of \(C\). The distribution is returned in the form of a sequence of tuples, where the \(i\)-th tuple contains the \(i\)-th weight, \(w_i\) say, and the number of codewords having weight \(w_i\).

Example: Weight Distribution of a Quadratic Residue Code (ex-601dad)

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In this example, the weight distribution of a quadratic residue code over \({\mathbb{Z}}_4\) and its dual are computed.

> C := QRCodeZ4(17);
> C;

((17, 4^9 2^0)) Cyclic Linear Code over IntegerRing(4)

Generator matrix:
[1 0 0 0 0 0 0 0 0 1 1 3 0 3 0 3 1]
[0 1 0 0 0 0 0 0 0 3 0 2 3 1 3 1 2]
[0 0 1 0 0 0 0 0 0 2 1 2 2 1 1 1 3]
[0 0 0 1 0 0 0 0 0 1 3 0 2 1 1 0 2]
[0 0 0 0 1 0 0 0 0 2 3 1 0 0 1 3 2]
[0 0 0 0 0 1 0 0 0 2 0 1 1 2 0 3 1]
[0 0 0 0 0 0 1 0 0 3 1 1 1 2 2 1 2]
[0 0 0 0 0 0 0 1 0 2 1 3 1 3 2 0 3]
[0 0 0 0 0 0 0 0 1 1 3 0 3 0 3 1 1]

> WeightDistribution(C);
[ <0, 1>, <5, 34>, <6, 68>, <7, 748>, <8, 2567>, <9, 6817>, <10, 17612>,
  <11, 34340>, <12, 50014>, <13, 56168>, <14, 50728>, <15, 30872>,
  <16, 9826>, <17, 2349> ]

> DualWeightDistribution(C);
[ <0, 1>, <6, 68>, <8, 935>, <9, 1632>, <10, 4148>, <11, 8568>, <12, 12886>,
  <13, 14280>, <14, 11968>, <15, 7752>, <16, 2890>, <17, 408> ]

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Lee Weight

For an element \(x\in{\mathbb{Z}}_4\), the Lee weight \(w_L(x)\) is defined by:

\[w_L(0) = 0,\quad w_L(1) = w_L(3) = 1,\quad w_L(2) = 2.\]

The Lee weight \(w_L(v)\) of a vector \(v\in{{\mathbb{Z}}_4^n}\) is defined to be the sum (in \({\mathbb{Z}}\)) of the Lee weights of its components. See [Wan97, p. 16].

LeeWeight(a): RngIntRes -> RngIntElt: The Lee weight of the element \(a\in{\mathbb{Z}}_4\).

LeeWeight(v): ModTupRngElt -> RngIntElt: The Lee weight of the codeword \(v\).

LeeDistance(u, v): ModTupRngElt, ModTupRngElt -> RngIntElt: The Lee distance between the codewords \(u\) and \(v\), where \(u\) and \(v\) belong to the same code \(C\). This is defined to be the Lee weight of \((u - v)\).

MinimumLeeWeight(C): Code -> RngIntElt
MinimumLeeDistance(C): Code -> RngIntElt: The minimum Lee weight of the code \(C\).

LeeWeightDistribution(C): Code -> SeqEnum: The Lee weight distribution of the code \(C\).

DualLeeWeightDistribution(C): Code -> SeqEnum: The Lee weight distribution of the dual of the code \(C\) (see LeeWeightDistribution)

WordsOfLeeWeight(C, w): Code, RngIntElt -> SetEnum

NumWords: RngIntElt                    Default: Infinity()

Given a linear code \(C\), return the set of all words of \(C\) having Lee weight \(w\). If NumWords is set to a non-negative integer \(c\), then the algorithm will terminate after a total of \(c\) words have been found.

WordsOfBoundedLeeWeight(C, l, u): Code, RngIntElt, RngIntElt -> SetEnum

NumWords: RngIntElt                    Default: Infinity()

Given a linear code \(C\), return the set of all words of \(C\) having Lee weight between \(l\) and \(u\), inclusive. If NumWords is set to a non-negative integer \(c\), then the algorithm will terminate after a total of \(c\) words have been found.

Example: Lee Weight Distribution (ex-587036)

We calculate the Lee weight distribution of a Reed Muller code over \({\mathbb{Z}}_4\) and enumerate all words of Lee weight \(8\).

> C := ReedMullerCodeZ4(1, 3);
> C;
(8, 256, 4) Linear Code over IntegerRing(4)
Generator matrix:
[1 0 0 0 3 1 2 1]
[0 1 0 0 2 1 1 3]
[0 0 1 0 1 1 3 2]
[0 0 0 1 3 2 3 3]
> LeeWeightDistribution(C);
[ <0, 1>, <6, 112>, <8, 30>, <10, 112>, <16, 1> ]
> W := WordsOfLeeWeight(C, 8);
> #W;
30

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Euclidean Weight

For an element \(x\in{\mathbb{Z}}_4\), the Euclidean weight \(w_E(x)\) is defined by:

\[w_E(0) = 0,\quad w_E(1) = w_E(3) = 1, \quad w_E(2) = 4.\]

The Euclidean weight \(w_E(v)\) of a vector \(v\in{{\mathbb{Z}}_4^n}\) is defined to be the sum (in \({\mathbb{Z}}\)) of the Euclidean weights of its components. See [Wan97, p. 16].

EuclideanWeight(a): RngIntRes -> RngIntElt: The Euclidean weight of the element \(a\in{\mathbb{Z}}4\).

EuclideanWeight(v): ModTupRngElt -> RngIntElt: The Euclidean weight of the \({\mathbb{Z}}_4\)-codeword \(v\).

EuclideanDistance(u, v): ModTupRngElt, ModTupRngElt -> RngIntElt: The Euclidean distance between the \({\mathbb{Z}}_4\)-codewords \(u\) and \(v\), where \(u\) and \(v\) belong to the same code \(C\). This is defined to be the Euclidean weight of \((u - v)\).

MinimumEuclideanWeight(C): Code -> RngIntElt
MinimumEuclideanDistance(C): Code -> RngIntElt: The minimum Euclidean weight of the \({\mathbb{Z}}_4\)-code C.

EuclideanWeightDistribution(C): Code -> SeqEnum: The Euclidean weight distribution of the \({\mathbb{Z}}_4\)-code C.

DualEuclideanWeightDistribution(C): Code -> SeqEnum: The Euclidean weight distribution of the dual of the \({\mathbb{Z}}_4\)-code C.

Example: Euclidean Weight Distribution (ex-9f8213)

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The Euclidean weight distribution is calculated for a quadratic residue code over \({\mathbb{Z}}_4\)

> C := QRCodeZ4(17);
> C;
(17, 262144) Cyclic Code over IntegerRing(4)
Generator matrix:
[1 0 0 0 0 0 0 0 0 1 1 3 0 3 0 3 1]
[0 1 0 0 0 0 0 0 0 3 0 2 3 1 3 1 2]
[0 0 1 0 0 0 0 0 0 2 1 2 2 1 1 1 3]
[0 0 0 1 0 0 0 0 0 1 3 0 2 1 1 0 2]
[0 0 0 0 1 0 0 0 0 2 3 1 0 0 1 3 2]
[0 0 0 0 0 1 0 0 0 2 0 1 1 2 0 3 1]
[0 0 0 0 0 0 1 0 0 3 1 1 1 2 2 1 2]
[0 0 0 0 0 0 0 1 0 2 1 3 1 3 2 0 3]
[0 0 0 0 0 0 0 0 1 1 3 0 3 0 3 1 1]
> EuclideanWeightDistribution(C);
[ <0, 1>, <7, 136>, <8, 170>, <9, 170>, <10, 408>, <11, 544>, <12, 986>,
<13, 1768>, <14, 3128>, <15, 5032>, <16, 6120>, <17, 6360>, <18, 8432>,
<19, 12512>, <20, 12682>, <21, 11152>, <22, 14416>, <23, 17680>, <24, 16048>,
<25, 15164>, <26, 17952>, <27, 16864>, <28, 13328>, <29, 14144>, <30, 14144>,
<31, 10064>, <32, 7837>, <33, 8024>, <34, 6800>, <35, 4896>, <36, 3485>,
<37, 2992>, <38, 2992>, <39, 1768>, <40, 510>, <41, 1258>, <42, 1224>,
<44, 238>, <45, 408>, <46, 136>, <47, 136>, <48, 34>, <68, 1> ]

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Weight Distributions#

Hamming Weight#

Lee Weight#

Euclidean Weight#

Weight Distributions

Hamming Weight

Lee Weight

Euclidean Weight