3.3 Weight Distribution and MacWilliams Identity

Weight distribution and MacWilliams identity are key concepts in linear codes. They help us understand how codewords are spread out and how errors affect transmission. These tools give us insights into a code's structure and performance.

The weight distribution counts codewords of each weight, while the MacWilliams identity links a code to its dual. Together, they reveal important properties like minimum distance and error-correcting capacity, without needing to examine every codeword.

Weight Distribution and Enumerator Polynomial

Defining Weight Distribution and Enumerator Polynomial

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• Weight distribution $A_i$ of a linear code $C$ counts the number of codewords of weight $i$ in $C$
  • Provides a statistical description of the code's structure and error-correcting capabilities
  • Can be used to compute the code's error probability and performance under various decoding algorithms
• Weight enumerator polynomial $A(z) = \sum_{i=0}^{n} A_i z^i$ is a generating function that encodes the weight distribution
  • Coefficients of $A(z)$ are the weight distribution values $A_i$
  • Allows for compact representation and algebraic manipulation of the weight distribution
  • Can be used to derive various properties of the code, such as its minimum distance and error-correcting capacity

Moment of Inertia and Its Applications

• Moment of inertia $I = \sum_{i=0}^{n} i A_i$ is the average weight of the codewords in $C$
  • Measures the concentration of codewords around the origin in the Hamming space
  • Higher moment of inertia indicates better error-correcting performance, as codewords are more spread out
• Moment of inertia can be computed from the weight enumerator polynomial $A(z)$ by evaluating its derivative at $z=1$
  • $I = A'(1) = \sum_{i=0}^{n} i A_i$
  • Provides a quick way to assess the code's error-correcting capabilities without computing the entire weight distribution

Symmetry Properties of Weight Distribution

• Weight distribution of a linear code is invariant under permutation of coordinates
  • Permuting the positions of the codeword bits does not change the weight distribution
  • Implies that the code's structure and properties are symmetric with respect to the coordinate positions
• Weight distribution is also invariant under translation by a codeword
  • Adding a codeword to all codewords in $C$ preserves the weight distribution
  • Reflects the linearity of the code and the group structure of the codeword space
• Symmetry properties can be used to simplify the computation and analysis of weight distributions
  • Allows for efficient enumeration of codewords and derivation of code properties
  • Can be exploited in the design of efficient decoding algorithms

MacWilliams Identity and Dual Codes

MacWilliams Identity and Its Implications

• MacWilliams identity relates the weight enumerator polynomial $A(z)$ of a linear code $C$ to that of its dual code $C^\perp$
  • $A^\perp(z) = \frac{1}{|C|} A(1-z, 1+z)$, where $A^\perp(z)$ is the weight enumerator of the dual code
  • Allows for the computation of the dual code's weight distribution from that of the original code
  • Implies a duality between the error-correcting properties of a code and its dual
• MacWilliams identity can be used to derive bounds on the minimum distance and error-correcting capacity of a code
  • Provides a way to assess the performance of a code without explicitly constructing its dual
  • Can be used to prove the optimality of certain codes, such as perfect codes and MDS codes

Dual Codes and Their Properties

• Dual code $C^\perp$ of a linear code $C$ is the set of vectors orthogonal to all codewords in $C$
  • $C^\perp = \{v \in \mathbb{F}_q^n : v \cdot c = 0 \text{ for all } c \in C\}$, where $\cdot$ denotes the inner product
  • Dual code is also a linear code, with dimension $n-k$ if $C$ has dimension $k$
  • Codewords in $C^\perp$ are the parity check equations for the codewords in $C$
• Properties of the dual code are closely related to those of the original code
  • Minimum distance of $C^\perp$ is equal to the minimum weight of the nonzero codewords in $C$
  • Generator matrix of $C^\perp$ is a parity check matrix for $C$, and vice versa
  • Encoding and decoding algorithms for $C$ can be adapted to work with $C^\perp$

Krawtchouk Polynomials and Invariance

• Krawtchouk polynomials $K_k(x;n)$ are a family of orthogonal polynomials that arise in the study of weight distributions
  • Defined by the recurrence relation $K_{k+1}(x;n) = (n-2x)K_k(x;n) - (n-k+1)K_{k-1}(x;n)$
  • Appear as coefficients in the MacWilliams identity, relating the weight enumerators of a code and its dual
  • Can be used to express various code properties, such as the weight distribution and the error probability
• Invariance properties of Krawtchouk polynomials reflect the symmetries of the Hamming space and the linear codes
  • $K_k(x;n) = (-1)^k K_k(n-x;n)$, reflecting the symmetry between the weights $x$ and $n-x$
  • $\sum_{x=0}^{n} K_k(x;n) K_j(x;n) = \delta_{kj} \binom{n}{k}$, expressing the orthogonality of the polynomials
  • Invariance properties can be used to simplify the computation and analysis of code properties