Symbol-Pair Metric Overview

• The symbol-pair metric is a distance measure defined on cyclic pair-read vectors that captures errors in overlapping adjacent symbol pairs, unlike traditional Hamming distance.
• It underpins advanced coding constructions, enabling the development of optimal MDS and AMDS codes through algebraic, cyclic, and repeated-root methodologies.
• The metric extends naturally to b-symbol metrics, influencing decoding strategies, covering theory, and even quantum CSS code designs for broader error correction applications.

The pair-symbol metric, more commonly called the symbol-pair metric, is a distance on words over a finite alphabet induced by a read model in which a channel outputs overlapping adjacent pairs rather than individual symbols. For a length-$n$ word over $\mathbb F_q$, the relevant observation is the cyclic pair-read vector $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$, and distance is measured by the Hamming distance between such pair-read vectors. The metric was proposed for symbol-pair read channels motivated by high-density storage systems, where isolated symbol reads may be physically impractical and pair-errors are more natural than ordinary symbol errors. In the subsequent literature, the pair-symbol metric became a central object for the construction of MDS and AMDS codes, for exact distance computation in cyclic and constacyclic families, and for extensions to the general $b$-symbol metric (Ding et al., 2016, Chen, 2022).

1. Formal definition and elementary structure

Let $x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$, with indices taken modulo $n$. The symbol-pair read vector is

$$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$

For $x,y\in \mathbb F_q^n$, the symbol-pair distance is

$$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$

equivalently the Hamming distance between $\pi(x)$ and $\mathbb F_q$0. The associated symbol-pair weight is

$\mathbb F_q$1

so that for linear codes $\mathbb F_q$2, and the minimum symbol-pair distance of a linear code is the minimum pair weight of its nonzero codewords (Tang et al., 2021, Ma et al., 2020).

The pair-symbol metric differs from the Hamming metric because it counts disagreements of adjacent ordered pairs rather than coordinatewise disagreements. A standard comparison used throughout the literature is

$\mathbb F_q$3

and similarly for linear codes,

$\mathbb F_q$4

A sharper structural statement is that if the support of the Hamming difference between $\mathbb F_q$5 and $\mathbb F_q$6 splits into $\mathbb F_q$7 consecutive blocks modulo $\mathbb F_q$8, then

$\mathbb F_q$9

This formula makes explicit that the pair-symbol metric is sensitive not only to the number of symbol disagreements but also to their local clustering (Ma et al., 2020, Tang et al., 2021).

Notation varies across the literature. The same distance appears as $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$0, $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$1, $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$2, or $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$3, and the corresponding weight as $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$4, $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$5, or $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$6. These are different notational realizations of the same pair-symbol metric (Ma et al., 2019).

2. Channel model, decoding meaning, and order sensitivity

The symbol-pair metric is tied to the symbol-pair read channel, where the decoder does not receive $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$7 itself but the cyclic list of adjacent pairs. This model was introduced to guard against pair-errors in high-density storage systems, where the read process may produce overlapping symbol pairs instead of isolated symbols. In that setting, the natural performance parameter is the minimum symbol-pair distance $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$8, not the ordinary Hamming distance (Ding et al., 2016, Ma, 2022).

As in classical coding theory, distance controls correction capability. The literature repeatedly uses the fact that a code with minimum pair distance $\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big)$9 can correct up to

$b$0

symbol-pair errors. This is the pair-metric analogue of the standard Hamming-metric decoding radius and explains why code constructions aim to maximize $b$1 rather than only $b$2 (Ma et al., 2020, Ma, 2022).

A common misconception is that pair distance is determined solely by Hamming weight. The literature shows instead that coordinate order matters. In the study of simplex and variation simplex codes, two permutations of the same simplex codeword were shown to have different symbol-pair weights: one had pair weight $b$3, another had pair weight $b$4. Thus rearranging coordinates can preserve Hamming weight while changing symbol-pair weight. This order sensitivity is one of the defining features of the pair-symbol metric and underlies several later construction methods based on coordinate permutations (Ma et al., 2019).

3. Bounds, optimality notions, and asymptotic length formulas

The basic upper bound is the Singleton-type bound for symbol-pair codes: $b$5 for a length-$b$6 $b$7-ary symbol-pair code with minimum pair distance $b$8. A code meeting this bound with equality is called an MDS symbol-pair code. For a linear $b$9 code, equality yields

$x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$0

Several papers also use the term AMDS symbol-pair code for the one-step-short case $x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$1, or equivalently for linear codes of size $x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$2 (Ma et al., 2020, Tang et al., 2022).

The same literature develops further extremal bounds in the generalized $x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$3-symbol setting. For

$x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$4

a Griesmer-type lower bound for linear $x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$5-symbol codes is

$x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$6

equivalently

$x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$7

For fixed $x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$8, the optimal length function satisfies

$x=(x_0,x_1,\dots,x_{n-1})\in \mathbb F_q^n$9

for all sufficiently large $n$0. In particular, for $n$1, this yields an eventual exact length formula for linear codes in the pair-symbol metric (Kurz, 10 Jul 2025).

Upper-bound phenomena also appear through covering theory. One paper proves that if a symbol-pair code has minimum pair distance $n$2, then for

$n$3

the code is not MDS. This is presented as a general upper bound on the lengths of MDS symbol-pair codes (Chen, 2022).

4. Exact distance computation and algebraic determination

A major strand of the theory concerns exact determination of pair distances for algebraically structured codes, especially repeated-root cyclic and constacyclic codes.

For cyclic codes of length $n$4 over $n$5, every code has the form

$n$6

The pair distances of all such codes were determined exactly. When $n$7,

$n$8

For $n$9, the complete classification is piecewise, with values such as $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$0, $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$1, $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$2, $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$3, $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$4, and $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$5 on explicitly described intervals of $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$6. This theorem gives the exact minimum symbol-pair distance for every repeated-root cyclic code of length $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$7 (Zhu et al., 2016).

For repeated-root cyclic codes over

$$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$8

of length $$\pi(x)=\big((x_0,x_1),(x_1,x_2),\dots,(x_{n-1},x_0)\big).$$9, the decisive invariant is the third torsional degree $x,y\in \mathbb F_q^n$0, defined via

$x,y\in \mathbb F_q^n$1

The paper shows that both the minimum symbol-pair weight and the minimum RT weight are determined from $x,y\in \mathbb F_q^n$2. In particular,

$x,y\in \mathbb F_q^n$3

while the minimum symbol-pair weight is a piecewise function of $x,y\in \mathbb F_q^n$4 indexed by the $x,y\in \mathbb F_q^n$5-adic position of $x,y\in \mathbb F_q^n$6. This reduces a pair-metric distance problem over a non-chain ring to a torsion-degree computation (Kim, 2020).

For reducible cyclic codes $x,y\in \mathbb F_q^n$7, cyclotomic numbers, generalized cyclotomic numbers, and Gaussian periods are used to determine possible symbol-pair weights through the count

$x,y\in \mathbb F_q^n$8

Under the stated $x,y\in \mathbb F_q^n$9 conditions, the minimum pair distance satisfies

$$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$0

The same work also identifies several three symbol-pair weight families and determines their symbol-pair weight distributions (Wang et al., 2023).

Framework	Structural mechanism	Pair-metric consequence
Cyclic codes of length $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$1	Generator $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$2	Exact classification of $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$3 for every $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$4
Repeated-root cyclic codes over $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$5	Third torsional degree $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$6	Piecewise formula for minimum pair weight; $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$7
Reducible cyclic codes $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$8	Cyclotomic numbers and Gaussian periods	Explicit pair-weight sets; in $$d_p(x,y)=\left|\left\{i\in \mathbb Z_n:(x_i,x_{i+1})\neq (y_i,y_{i+1})\right\}\right|,$$9 cases $\pi(x)$0

These results show that the pair-symbol metric is not merely an auxiliary reformulation of Hamming distance. In the algebraic code families where exact analysis is possible, pair distance exhibits its own piecewise regimes, its own extremal constructions, and its own governing invariants.

5. Explicit construction of MDS and AMDS symbol-pair codes

The construction theory of optimal pair-symbol codes is extensive and uses projective geometry, elliptic curves, repeated-root cyclic codes, constacyclic codes, matrix-product codes, and simple-root cyclic codes.

A foundational geometric construction proves that a linear MDS $\pi(x)$1 symbol-pair code exists if and only if

$\pi(x)$2

For pair distance $\pi(x)$3, linear MDS $\pi(x)$4 symbol-pair codes exist for all

$\pi(x)$5

using ovoids in $\pi(x)$6. With elliptic curves, linear MDS $\pi(x)$7 symbol-pair codes are constructed for

$\pi(x)$8

where $\pi(x)$9 or $\mathbb F_q$00 (Ding et al., 2016).

Repeated-root cyclic constructions produce high pair distances. For prime $\mathbb F_q$01 with $\mathbb F_q$02, MDS $\mathbb F_q$03 and $\mathbb F_q$04 symbol-pair codes are constructed, and for every odd prime $\mathbb F_q$05 there is an MDS $\mathbb F_q$06 symbol-pair code (Ma et al., 2020). Other repeated-root constructions over $\mathbb F_q$07 with $\mathbb F_q$08 give MDS $\mathbb F_q$09 and $\mathbb F_q$10 symbol-pair codes (Ma et al., 2020). A later paper constructs six AMDS families with

$\mathbb F_q$11

including one class of unbounded lengths and one class with minimum symbol-pair distance $\mathbb F_q$12 (Ma, 2022).

Constacyclic classification results identify exactly which codes in certain repeated-root constacyclic classes are MDS. Over $\mathbb F_q$13, the listed MDS families include $\mathbb F_q$14, $\mathbb F_q$15, $\mathbb F_q$16 in the special case $\mathbb F_q$17, $\mathbb F_q$18 for $\mathbb F_q$19, $\mathbb F_q$20, $\mathbb F_q$21, $\mathbb F_q$22 for $\mathbb F_q$23, and $\mathbb F_q$24. The same work proves there are no other nontrivial MDS symbol-pair constacyclic codes in the studied classes over finite fields and the ring $\mathbb F_q$25 with $\mathbb F_q$26 (Tang et al., 2021).

Matrix-product methods with permutation equivalence produce six further MDS classes. Under the stated congruence conditions, the constructions yield MDS $\mathbb F_q$27, $\mathbb F_q$28, $\mathbb F_q$29, $\mathbb F_q$30, $\mathbb F_q$31, and $\mathbb F_q$32 symbol-pair codes over prime power fields. The key mechanism is that permutation equivalence preserves Hamming parameters but can enlarge symbol-pair distance by changing adjacency patterns (Zheng et al., 2024).

Simple-root cyclic constructions extend the known range of long optimal codes. Three infinite families are obtained: $\mathbb F_q$33

$\mathbb F_q$34

and

$\mathbb F_q$35

The same paper states that for pair distance $\mathbb F_q$36 or $\mathbb F_q$37, these $\mathbb F_q$38-ary MDS symbol-pair codes achieve the longest known code length when $\mathbb F_q$39 is not a prime, and for pair distance $\mathbb F_q$40, they attain the longest known code length regardless of whether $\mathbb F_q$41 is prime (Qiu et al., 26 Mar 2025).

6. Generalizations, distributions, covering theory, and quantum extensions

The pair-symbol metric is the case $\mathbb F_q$42 of the $\mathbb F_q$43-symbol metric. For

$\mathbb F_q$44

the $\mathbb F_q$45-symbol weight and distance are

$\mathbb F_q$46

The standard comparison becomes

$\mathbb F_q$47

This generalization is central in covering theory, list decoding, and asymptotic length questions (Chen, 2022, Kurz, 10 Jul 2025).

Weight distribution theory reveals further metric-specific behavior. For $\mathbb F_q$48 MDS codes, the symbol-pair weight distribution $\mathbb F_q$49 satisfies

$\mathbb F_q$50

and explicit formulas are derived for $\mathbb F_q$51. For cyclic simplex codes, all nonzero codewords have the same symbol-$\mathbb F_q$52-weight: $\mathbb F_q$53 In particular, in the pair-symbol case $\mathbb F_q$54, every nonzero cyclic simplex codeword has

$\mathbb F_q$55

Variation simplex codes exhibit a different uniform value,

$\mathbb F_q$56

while also illustrating that coordinate rearrangement can change pair weight (Ma et al., 2019).

Covering theory in the pair and $\mathbb F_q$57-symbol metrics diverges from its Hamming counterpart. The literature proves that there is no perfect linear symbol-pair code with minimum pair distance $\mathbb F_q$58 and no perfect $\mathbb F_q$59-symbol metric code if

$\mathbb F_q$60

It also determines the covering radius of Reed–Solomon codes: $\mathbb F_q$61 These results feed into generalized Singleton-type bounds for list-decodable $\mathbb F_q$62-symbol codes (Chen, 2022).

A recent quantum extension relates symbol-pair weight to symplectic weight for CSS decoding. For a binary vector $\mathbb F_q$63 and its cyclic left shift $\mathbb F_q$64,

$\mathbb F_q$65

Using this identity, a symbol-pair syndrome decoder for classical cyclic codes is lifted to a decoder for CSS codes, yielding improved correction capability for certain cyclic CSS constructions (Jha et al., 2024).

Taken together, these developments show that the pair-symbol metric has evolved from a storage-motivated distance on overlapping pairs into a mature metric framework with its own extremal theory, exact algebraic classifications, covering and list-decoding bounds, metric-specific weight distributions, and even quantum-decoding applications.