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Generator matrix
Generator matrix
Generator matrix
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In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix.

Terminology

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If G is a matrix, it generates the codewords of a linear code C by

where w is a codeword of the linear code C, and s is any input vector. Both w and s are assumed to be row vectors.[1] A generator matrix for a linear -code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.). The number of redundant bits is denoted by .

The standard form for a generator matrix is,[2]

,

where is the identity matrix and P is a matrix. When the generator matrix is in standard form, the code C is systematic in its first k coordinate positions.[3]

A generator matrix can be used to construct the parity check matrix for a code (and vice versa). If the generator matrix G is in standard form, , then the parity check matrix for C is[4]

,

where is the transpose of the matrix . This is a consequence of the fact that a parity check matrix of is a generator matrix of the dual code .

G is a matrix, while H is a matrix.

Equivalent codes

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Codes C1 and C2 are equivalent (denoted C1 ~ C2) if one code can be obtained from the other via the following two transformations:[5]

  1. arbitrarily permute the components, and
  2. independently scale by a non-zero element any components.

Equivalent codes have the same minimum distance.

The generator matrices of equivalent codes can be obtained from one another via the following elementary operations:[6]

  1. permute rows
  2. scale rows by a nonzero scalar
  3. add rows to other rows
  4. permute columns, and
  5. scale columns by a nonzero scalar.

Thus, we can perform Gaussian elimination on G. Indeed, this allows us to assume that the generator matrix is in the standard form. More precisely, for any matrix G we can find an invertible matrix U such that , where G and generate equivalent codes.

See also

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Notes

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References

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Further reading

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Generator matrix

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In , a generator matrix is a k×nk \times n matrix GG over a that defines an [n,k][n, k] linear , where the codewords are all possible linear combinations of its rows, enabling the systematic encoding of kk-symbol messages into nn-symbol codewords via c=mGc = mG. The rows of GG must be linearly independent to span the code space exactly, ensuring the code has dimension kk, and the matrix fully characterizes the linear code's structure, with codewords forming a kk-dimensional subspace of the nn-dimensional over the field. This representation allows for efficient encoding, requiring O(nk)O(nk) operations, and supports the construction of various error-correcting codes, such as Hamming codes, by specifying parity bits derived from message symbols. A particularly useful form is the systematic generator matrix, where G=[Ik∣P]G = [I_k \mid P], with IkI_k the k×kk \times k and PP a k×(n−k)k \times (n-k) matrix; this ensures the first kk positions of each codeword directly match the message bits, while the remaining n−kn-k positions hold parity checks, simplifying encoding and decoding processes. Any generator matrix can be transformed into this systematic form through elementary row and column operations without altering the , though the resulting matrix is not unique for a given . The generator matrix is intimately related to the parity-check matrix HH, an (n−k)×n(n-k) \times n matrix such that GHT=0GH^T = 0, meaning all codewords satisfy cHT=0cH^T = 0; this allows HH to detect and correct errors via computation during decoding, while GG handles the forward encoding. In applications like data transmission over noisy channels, the choice of GG influences the code's , which determines error-correcting capability—for instance, ensuring no zero or linearly dependent columns in the corresponding HH guarantees single-error detection. Generator matrices extend to cyclic codes through generator polynomials and find use in modern systems for reliable communication and storage.

Fundamentals

Definition

In coding theory, a generator matrix GG for a linear block code CC is defined as a k×nk \times n matrix whose entries are elements of a Fq\mathbb{F}_q (also denoted GF(q)), where qq is a , and whose rows form a basis for the kk-dimensional subspace CC of the Fqn\mathbb{F}_q^n. This basis ensures that the rows of GG are linearly independent and span the entire code subspace, with the rank of GG equal to kk. The codewords of CC are generated as all possible linear combinations of the rows of GG; specifically, every codeword c∈C\mathbf{c} \in C can be expressed as c=uG\mathbf{c} = \mathbf{u} G, where u\mathbf{u} is a 1×k1 \times k (row) vector, known as the vector, with entries in Fq\mathbb{F}_q. Here, the kk of the code corresponds to the number of symbols (or rows of GG), while the code length nn is the total number of symbols in each codeword (or columns of GG), introducing a redundancy of n−kn - k symbols to enable . This construction assumes familiarity with basic linear algebra over finite fields, including vector spaces and subspace bases. The generator matrix complements the parity-check matrix, which defines the of CC.

Basic Properties

A generator matrix GG for a linear [n,k,d]q[n, k, d]_q code CC is a k×nk \times n matrix over the Fq\mathbb{F}_q that has full row rank kk. This full row rank property guarantees that the rows of GG are linearly independent and span the kk-dimensional subspace CC. The row space of GG is precisely the code CC, consisting of all possible linear combinations of its rows. As a result, the of CC is qkq^k, reflecting the dimension of the subspace over Fq\mathbb{F}_q. Any two generator matrices generate the same code CC if and only if their rows form different bases for the same row space. A particularly convenient representation is the systematic form of GG, where the first kk columns form the k×kk \times k identity matrix. The minimum Hamming distance dd of CC equals the minimum Hamming weight among all nonzero codewords, which corresponds to the minimum weight of any nonzero linear combination uGuG for u∈Fqk∖{0}u \in \mathbb{F}_q^k \setminus \{0\}. Thus, every nonzero codeword c=uGc = uG satisfies wt(c)≥d.\mathrm{wt}(c) \geq d.

Standard Forms and Construction

Systematic Form

In coding theory, the systematic form of a generator matrix provides a standardized representation for linear block codes that facilitates encoding and analysis. For an (n, k) linear block code over a finite field, the generator matrix G in systematic form is a k × n matrix structured as G = [I_k | P], where I_k denotes the k × k identity matrix and P is a k × (n - k) matrix referred to as the parity submatrix. This arrangement ensures that the rows of G form a basis for the code, with the identity block directly embedding the information positions. A key advantage of this form lies in its direct mapping from the message to the codeword. Given a message vector u of length k, the corresponding codeword c = u G takes the explicit form c = [u | u P], where the first k symbols of c are identical to u (the information symbols), and the remaining n - k symbols are linear combinations determined by P (the parity symbols). This structure simplifies processes, as the original message can be recovered immediately from the codeword without additional decoding steps in the absence of errors. Any full-rank generator matrix can be converted to systematic form through elementary row operations, such as row additions and swaps, which preserve the row space and thus the underlying code. The transformation process employs to reduce the matrix until the leading k × k submatrix becomes the identity, ensuring the code's remains k. This equivalence holds because row operations correspond to linear combinations of codewords, maintaining the code's properties without altering its minimum distance or error-correcting capability.

General Construction Methods

One fundamental method to construct a generator matrix GG for an [n,k][n, k] linear block code CC over a Fq\mathbb{F}_q involves selecting a set of kk linearly independent codewords that span CC. These codewords are arranged as the rows of GG, ensuring the row space equals CC by verifying linear independence (e.g., via the rank of GG being kk) and that all codewords can be expressed as linear combinations of these rows. This direct basis selection is applicable when the codewords are known or can be generated from the code definition, providing a straightforward way to define GG without additional structure. For cyclic codes, a specific construction leverages the generator polynomial g(x)g(x) of degree n−kn-k. The generator matrix GG is built as a k×nk \times n matrix where the first row consists of the coefficients of g(x)g(x) followed by k−1k-1 zeros, and each subsequent row is a right cyclic shift of the row above it. This circulant structure ensures the rows generate all cyclic shifts of multiples of g(x)g(x), spanning the code. Such matrices often yield non-systematic forms, though row operations can transform them into systematic ones if desired. Constructions considering the Singleton bound, which states d≤n−k+1d \leq n - k + 1 for minimum distance dd, aim for maximum distance separable (MDS) codes achieving equality. Reed-Solomon codes exemplify this: over Fq\mathbb{F}_q with n≤qn \leq q, the k×nk \times n generator matrix GG has entries Gi,j=α(i−1)(j−1)G_{i,j} = \alpha^{(i-1)(j-1)} for i=1,…,ki = 1, \dots, k, j=1,…,nj = 1, \dots, n, where α\alpha is a primitive element; evaluation at distinct points (powers of α\alpha) ensures any kk columns are linearly independent, guaranteeing the MDS property. Algorithmically, a basis for GG can be found using linear algebra techniques on a generating set of codewords, such as row reduction via to extract kk independent rows, or leveraging computations to identify vectors in the code subspace, though these methods focus on conceptual extraction without detailed steps. For illustration, in a binary [7,4][7,4] code, one might select four linearly independent codewords (e.g., from the code's defining relations) as rows, verifying the span covers all 24=162^4 = 16 codewords to confirm GG's validity.

Relation to Other Matrices

Parity-Check Matrix

In , the parity-check matrix HH of a linear CC over a Fq\mathbb{F}_q with length nn and dimension kk is defined as an (n−k)×n(n-k) \times n matrix of full row rank such that cHT=0\mathbf{c} H^T = \mathbf{0} for every codeword c∈C\mathbf{c} \in C./08:_Algebraic_Coding_Theory/8.04:_Parity-Check_and_Generator_Matrices) This condition ensures that the rows of HH span the of the code subspace, providing a set of linear equations that all codewords satisfy. The parity-check matrix HH is orthogonal to the generator matrix GG of the code, satisfying GHT=0k×(n−k)G H^T = \mathbf{0}_{k \times (n-k)}, where 0\mathbf{0} denotes the ./08:_Algebraic_Coding_Theory/8.04:_Parity-Check_and_Generator_Matrices) This implies that the row space of GG, which generates CC, lies in the null space of HH, and vice versa, the rows of HH form a basis for the of vectors orthogonal to every codeword. Note that HH is itself a generator matrix for the dual code C⊥C^\perp. For a generator matrix in systematic form G=[Ik∣P]G = [I_k \mid P], where IkI_k is the k×kk \times k identity matrix and PP is a k×(n−k)k \times (n-k) matrix, the corresponding parity-check matrix is H=[−PT∣In−k]H = [-P^T \mid I_{n-k}]./08:_Algebraic_Coding_Theory/8.04:_Parity-Check_and_Generator_Matrices) Over F2\mathbb{F}_2, the negation is irrelevant since −1=1-1 = 1, yielding H=[PT∣In−k]H = [P^T \mid I_{n-k}]. This form directly encodes the parity relations from the systematic structure. The syndrome computation uses HH for error detection: for a received vector r=c+e\mathbf{r} = \mathbf{c} + \mathbf{e}, the syndrome is s=rHT=eHT\mathbf{s} = \mathbf{r} H^T = \mathbf{e} H^T, which equals 0\mathbf{0} if and only if r\mathbf{r} is a codeword (no detectable error). Nonzero syndromes indicate the presence of errors and can guide decoding by matching to possible error patterns. To derive HH from a general generator matrix GG, compute a basis for the null space of GG, which consists of all vectors orthogonal to the rows of GG; this can be achieved through on an augmented form of GG to identify the . The resulting basis vectors, arranged as rows, form HH, ensuring full rank n−kn-k./08:_Algebraic_Coding_Theory/8.04:_Parity-Check_and_Generator_Matrices)

Dual Code Generator

In coding theory, the dual code C⊥C^\perp of a linear block code C⊆FqnC \subseteq \mathbb{F}_q^n of length nn and dimension kk is defined as the set of all vectors z∈Fqnz \in \mathbb{F}_q^n such that the standard dot product ⟨z,c⟩=∑i=1nzici=0\langle z, c \rangle = \sum_{i=1}^n z_i c_i = 0 for every codeword c∈Cc \in C. This dual code C⊥C^\perp is itself a linear code of dimension n−kn - k. The orthogonality condition ensures that C⊥C^\perp captures the vectors perpendicular to the entire subspace spanned by CC. The generator matrix G⊥G^\perp for the dual code C⊥C^\perp is a (n−k)×n(n-k) \times n matrix whose rows form a basis for C⊥C^\perp. If HH is a full-rank parity-check matrix for the original code CC, then G⊥G^\perp can be taken as HH (up to row permutations or scaling for equivalence). This relation arises because the rows of HH are orthogonal to every codeword in CC, satisfying the defining property of C⊥C^\perp. Specifically, if GG is the generator matrix for CC, the orthogonality is expressed by the equation HGT=0(n−k)×k,H G^T = 0_{(n-k) \times k}, which holds over Fq\mathbb{F}_q and confirms that the rows of HH lie in the . A key property of linear codes is the double dual relation: the dual of the dual code recovers the original, i.e., (C⊥)⊥=C(C^\perp)^\perp = C. This follows from the dimensions adding to nn and the fact that CC is contained in (C⊥)⊥(C^\perp)^\perp, with equality in finite fields. Regarding error-correcting capabilities, the minimum distance of the dual code relates to that of the original in specific families, such as through adaptations of the BCH bound for BCH codes and their duals, though exact relations depend on the code structure.

Encoding and Code Generation

Systematic Encoding

In systematic encoding, a message vector u∈Fqk\mathbf{u} \in \mathbb{F}_q^k (typically treated as a row vector) is mapped to a codeword c∈Fqn\mathbf{c} \in \mathbb{F}_q^n using the systematic generator matrix G=[Ik∣P]G = [I_k \mid P], where IkI_k is the k×kk \times k and PP is a k×(n−k)k \times (n-k) matrix over the Fq\mathbb{F}_q. The resulting codeword takes the form c=uG=[u∣uP]\mathbf{c} = \mathbf{u} G = [\mathbf{u} \mid \mathbf{u} P], directly embedding the kk message symbols as the first kk positions of c\mathbf{c}, while the remaining n−kn-k parity symbols are linear combinations determined by PP. This structure offers key advantages, including the ability to extract the original information symbols from a received codeword by simply selecting the first kk positions, which reduces complexity in applications requiring partial decoding or verification. The encoding process has a of O(kn)O(kn) operations over Fq\mathbb{F}_q, dominated by the parity computation, though the direct copying of message bits contributes negligibly. Implementations in hardware or software typically involve straightforward matrix-vector over Fq\mathbb{F}_q, with optimizations such as precomputed lookup tables for the parity submatrix PP in cases of small field sizes or block lengths to accelerate the linear combinations. The parity symbols are computed exclusively as functions of the input message u\mathbf{u}, ensuring a process with no dependence on intermediate results or feedback loops. Explicitly, the codeword components are given by ci=uifor i=1,…,k,c_i = u_i \quad \text{for } i = 1, \dots, k, and ck+j=∑i=1kuiPijc_{k+j} = \sum_{i=1}^k u_i P_{i j} (where the sum is over field operations in Fq\mathbb{F}_q) for j=1,…,n−kj = 1, \dots, n-k.
This formulation holds for any linear admitting a systematic generator matrix, which is always possible via standard form transformations.

Non-Systematic Encoding

In non-systematic encoding, a codeword c\mathbf{c} is generated by multiplying the message vector u∈Fqk\mathbf{u} \in \mathbb{F}_q^k by a full-rank generator matrix G∈Fqk×n\mathbf{G} \in \mathbb{F}_q^{k \times n} that lacks an embedded k×kk \times k identity submatrix, yielding c=uG\mathbf{c} = \mathbf{u} \mathbf{G}. This process computes each component of c\mathbf{c} as a of the message symbols weighted by the corresponding column of G\mathbf{G}, ensuring all codewords are produced without direct extraction of the message bits from the codeword. Equivalently, the encoding can be expressed as c=∑i=1kuigi,\mathbf{c} = \sum_{i=1}^k u_i \mathbf{g}_i, where gi\mathbf{g}_i denotes the ii-th row of G\mathbf{G}, highlighting the codeword as a weighted sum of the basis vectors spanning the code subspace. The of this direct is O(kn)O(kn) field operations, which can become prohibitive for large nn without exploiting any inherent structure in G\mathbf{G}. To mitigate encoding inefficiency, a non-systematic G\mathbf{G} can be temporarily transformed into systematic form via (row reduction to ) and column permutations, allowing encoding in the equivalent systematic code before optionally reverting if needed; however, the resulting code remains unchanged up to equivalence. Non-systematic encoding is particularly employed in constructions like cyclic codes, where the natural generator matrix derived from the generator polynomial is non-systematic and preserves the cyclic shift property essential for efficient syndrome computation and decoding, whereas a systematic form might disrupt this structure.

Code Equivalence

Permutation Equivalence

In , two linear codes over a are permutation equivalent if one can be obtained from the other by applying a to the coordinate positions, meaning there exists a on the set of coordinates that maps the codewords of the first code onto those of the second. For generator matrices, this equivalence corresponds to transforming a generator matrix GG of size k×nk \times n for code CC into a new generator matrix G′=GΠG' = G \Pi, where Π\Pi is an n×nn \times n permutation matrix representing the coordinate permutation. The resulting code C′C' generated by G′G' has codewords c′=cΠc' = c \Pi for each codeword cc in CC, ensuring that every codeword in C′C' is a permuted version of a codeword in CC. This transformation preserves key structural properties of the , including its nn, kk, and minimum dd, since permuting coordinates does not alter the number of nonzero entries in any codeword or the pairwise distances between them. The weight distribution, which counts the number of codewords of each possible , remains unchanged as a , though the specific positions of weights within individual codewords are reordered. Consequently, invariants such as the weight enumerator are identical for permutation-equivalent codes. Permutation equivalence finds practical application in simplifying the structural analysis of linear codes or in reordering coordinates to obtain a more convenient representation, such as facilitating the identification of systematic forms without altering the underlying code properties. For example, by selecting an appropriate permutation matrix Π\Pi and applying it to the columns of GG, one can rearrange the generator matrix to highlight certain patterns in the code, which aids in encoding processes or comparative studies of code performance. This operation is a foundational tool in code classification, as it allows researchers to normalize codes up to position permutations before deeper equivalence checks. Permutation equivalence serves as a building block for broader notions like monomial equivalence, which extends it by incorporating field element scalings on coordinates.

Monomial Equivalence

Monomial equivalence provides a broader notion of code similarity than permutation equivalence alone, incorporating both rearrangements of positions and scalings of symbols within the . Two linear s over Fq\mathbb{F}_q with generator matrices GG and G′G', both of size k×nk \times n, are monomially equivalent if there exists an n×nn \times n Π\Pi and an n×nn \times n Δ\Delta with nonzero diagonal entries from Fq\mathbb{F}_q such that G′=GΠΔG' = G \Pi \Delta. This transformation preserves the of the , its kk, and the minimum , as the column scalings by nonzero elements of Δ\Delta preserve the positions of zeros (and thus Hamming weights and distances), while Π\Pi permutes the columns corresponding to position relabeling. In the context of codes over the Fq\mathbb{F}_q, the scalings are performed by multiplying entries in specific positions (columns) by nonzero elements of Fq\mathbb{F}_q, effectively relabeling the field symbols while maintaining the code's structural properties. The transformed codewords satisfy c′=cΠΔc' = c \Pi \Delta, where each diagonal entry of Δ\Delta is in Fq×\mathbb{F}_q^\times, reflecting the combined effect of and per-position scaling; this aligns with the generator matrix transformation G′=GΠΔG' = G \Pi \Delta, as message vectors mm yield mG′=(mG)ΠΔ=cΠΔm G' = (m G) \Pi \Delta = c \Pi \Delta. This equivalence relates closely to the groups of codes, where group actions—generated by permutations and diagonal scalings—define the symmetries that classify codes . Specifically, the orbits under the group partition the of linear codes into equivalence classes, enabling the determination of whether two codes are essentially identical modulo relabeling and position permutations, a key tool in code classification and . Permutation equivalence arises as the special case where the diagonal scalings are trivial (i.e., all diagonal entries of Δ\Delta are 1).

Examples

Binary Hamming Code

The binary is a [7,4,3] binary linear , meaning it has n=7n=7, k=4k=4, and minimum distance d=3d=3, which allows for single-error correction. This , introduced by , encodes 4 information bits into 7 bits by adding 3 parity bits, forming a perfect that achieves the for single-error correction over the binary field. A standard systematic generator matrix GG for the binary Hamming code takes the form G=[I4∣P]G = [I_4 \mid P], where I4I_4 is the 4×4 and PP is a 4×3 parity submatrix derived from the parity-check requirements. The explicit matrix is G=(1000110010010100100110001111),G = \begin{pmatrix} 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{pmatrix},
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