Description
Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), i.e., \(q\)-ary strings. Error-correcting performance is quantified by some distance \(d\), which in turn is defined using a metric. The default distance is the Hamming distance \(d\), the weight (i.e., number of nonzero coordinates) of the lowest-weight nonzero codeword; such codes are usually denoted as \((n,K,d)_q\). The corresponding Hamming metric between two \(q\)-ary strings is the number of coordinates in which they differ. Unless stated otherwise, the distance for this class is the Hamming distance.
Two \(q\)-ary codes are equivalent if the codewords of one code can be mapped into those of the other under a combination of a coordinate permutation and a permutation of the elements of each coordinate. The full group of such composite permutations is \(S_q \wr S_n\) [2][1; Def. 1.8.8].
Protection
A code detects errors on up to \(d-1\) coordinates, corrects erasure errors on up to \(d-1\) coordinates, and corrects general errors on up to \(\left\lfloor (d-1)/2 \right\rfloor\) coordinates. Often, the relative Hamming distance \(\delta=d/n\) is used to compare codes of different lengths.
An analogue of the Gray map and Lee weight can be defined for codes over \(GF(4)\) by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [3].
Noise channels
Noise channels include the symmetric noise channel, asymmetric noise channels [4–8], as well as insertion/deletion noise and its generalization to substring edits [9].
Weight enumerator and four fundamental parameters
Weight enumerator: Determining protection and bounds on code parameters can also be done using the code's weight enumerator (cf. quantum weight enumerators), \begin{align} \begin{split} A(x,y)&=\sum_{j=0}^{n}A_{j}x^{n-j}y^{j}\\ A_{j}&=\text{number of wt-}j\text{ codewords}~. \end{split} \tag*{(1)}\end{align} The weight enumerator and it's Fourier transform the dual weight enumerator satisfy the MacWilliams identity [10,11]; see [12; Ch. 5].
The distance of the code is the value of the first nonzero coefficient \(A_j\), while the dual distance is the first nonzero coefficient of the dual weight enumerator. The number of the code is the number of nonzero \(A_j\)'s, corresponding to the number of distinct nonzero distances between codewords. The external distance is the number of nonzero coefficients of the dual weight enumerator. The distance, dual distance, number and external distance make up the four fundamental parameters of a code [13][12; Ch. 5].
Other types of weight enumerators includes the Hamming weight enumerator, Lee weight enumerator, joint weight enumerator, split weight enumerator, and biweight enumerator [12].
Bounds on code parameters
Bounds on the parameters of an \((n,K,d)_q\) code include the Hamming a.k.a. sphere-packing bound, Singleton bound, Gilbert-Varshamov (GV) bound, Griesmer bound, Plotkin bound, Johnson bound, and various linear programming (LP) bounds; see [1]. A code whose parameters attain the Hamming bound (Singleton bound, Griesmer bound, Johnson bound, Delsarte LP bound) is called a perfect code (an MDS code, a Griesmer code, a nearly perfect code, an LP universally optimal code).
Gilbert-Varshamov (GV) bound: The Gilbert-Varshamov [14,15], or Gilbert-Shannon-Varshamov, bound states that the maximum size \(K\) of a \(q\)-ary code with distance \(d\) satisfies \begin{align} K\sum_{j=0}^{d-1}{n \choose j}(q-1)^{j}\geq q^{n}~. \tag*{(2)}\end{align} In other words, if the left-hand side of the above is less than or equal to the right-hand side, then a code with such parameters exists. The GV bound gives rise to the asymptotic GV bound (i.e., GV bound in the \(n\to\infty\) limit), expressed in terms of the maximum achievable rate \(R\) and relative distance \(\delta\), \begin{align} R\geq 1-h_{q}(\delta)~, \tag*{(3)}\end{align} where \(h_q\) is the \(q\)-ary entropy function, \begin{align} h_{q}(\delta)=-\delta\log_{q}\frac{\delta}{q-1}-(1-\delta)\log_{q}(1-\delta)~. \tag*{(4)}\end{align}
Rate
The rate of a \(q\)-ary code is usually defined as \(R=\frac{1}{n}\log_q K\) dits per symbol. The maximum rate of a \(q\)-ary code with linear distance is lower bounded by the asymptotic GV bound and upper bounded by the \(q\)-ary version of the MRRW LP bound [16].Decoding
For small \(n\), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\). The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\).Threshold
Threshold for large-alphabet circuits codes is higher than for Boolean circuits [17].Notes
Tables of bounds and examples of linear codes for various \(n\) and \(k\), extending code tables by Brouwer [18], are maintained by M. Grassl at this website.Cousins
- Matrix-based code— Elements of fields such as \(GF(p^{ml})\) can be written as \(m\)-dimensional vectors over \(GF(p^l)\) or \((m\times l)\)-dimensional matrices over \(GF(p)\). This idea is used to convert between ordinary block codes and matrix-based codes such as disk array codes and rank-metric codes.
- Combinatorial design— Designs can be constructed from \(q\)-ary codes by taking the supports of a subset of codewords of constant weight.
- Constantin-Rao (CR) code— CR codes, and their special cases the VT codes, can be converted to ternary codes with nice structure via a binary-to-ternary map \(00\to 0\), \(11\to 0\), \(01\to 1\), and \(10\to 2\) [19].
- Traceability code— A \(q\)-ary code with distance \(d \geq n(1-1/t^2)\) has the \(t\)-traceability property [20; Thm. 4.3].
- Convolutional code— Convolutional codes for finite block size are \(q\)-ary codes.
- Polyphase code— Polyphase codes are spherical codes that can be obtained from \(q\)-ary codes.
Primary Hierarchy
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Page edit log
- Victor V. Albert (2022-02-16) — most recent
- Victor V. Albert (2021-10-29)
Cite as:
“\(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_digits_into_q-ary_digits