Quotient-Based Decoding Method

• Quotient-based decoding is an algebraic error-correction method that partitions the ambient space into cosets using quotient structures like rings, semirings, and lattices.
• It employs syndrome maps to compute coset identities and selects canonical minimal-weight coset leaders for effective error correction.
• This approach underpins diverse applications, including classical linear, Reed-Muller, skew-BCH, and Ternary Gamma-Semiring (TGS) codes, balancing precomputation with polynomial online decoding.

A quotient-based decoding method is an algebraic error-correction procedure in which the ambient space is partitioned into cosets (syndrome classes) via a quotient structure—typically a quotient ring, semiring, ideal lattice, or module. Codewords form one distinguished coset (the kernel), and decoding proceeds by associating a received word to its coset (via a syndrome computation) and selecting a canonical representative (coset leader) for error correction. The method applies to a range of algebraic frameworks, notably classical linear codes, codes over noncommutative rings, and, fundamentally, to codes constructed from the ideal lattices of finite commutative ternary Gamma-semirings (TGS), as well as to quotient variants of Reed-Muller codes and certain convolutional and algebraic-geometric codes (Gokavarapu et al., 24 Nov 2025, Gotlib et al., 21 Feb 2025, Gómez-Torrecillas et al., 2016, Andriatahiny et al., 2018).

1. Foundational Algebraic Frameworks

Quotient-based decoding leverages quotient constructions from algebra, where an ambient set (frequently a module or direct power) is partitioned via an equivalence relation determined by an ideal, submodule, or subgroup. In classical binary linear coding, this amounts to the coset decomposition $\mathbb{F}_q^n / C$ for a code $C$. However, recent generalizations include:

• Ternary Gamma-Semirings: A finite commutative ternary Gamma-semiring $(T, \oplus, [\cdot,\cdot,\cdot], \Gamma)$, where the fundamental operations are a commutative, idempotent binary $\oplus$ and a commutative, associative, distributive ternary operation $[x,y,z]$, endowed with a commutative monoid action $\Gamma$ compatible in all slots (Gokavarapu et al., 24 Nov 2025).
• Ideal Lattices: The substructure of $k$-ideals (subsets closed under the defined operations and suitably downward closed) yields a distributive lattice $L(T)$. The quotient set $T/I$ consists of absorption classes under the ideal $I$.
• Quotient Polynomial and Skew Rings: Skew BCH convolutional codes use quotients of noncommutative skew-polynomial rings, while Reed-Muller quotient codes restrict evaluation to subvarieties or subsets of the ambient space (Gotlib et al., 21 Feb 2025, Gómez-Torrecillas et al., 2016).

In all cases, a quotient structure partitions the word space, and the decoding process fundamentally relies on identifying and exploiting these coset structures.

2. Syndrome Maps and Coset Partitioning

Syndrome-based decoding in quotient frameworks generalizes as follows:

• Syndrome Map: An algebraic map $\Phi$ from the ambient word space to the base algebra (e.g., $\Phi: T^n \rightarrow T$ or $\mathbb{F}_q^n \rightarrow \mathbb{F}_q^m$) is constructed so that $\ker\,\Phi$ is the code. For TGS-codes, $\Phi$ is typically built from sums of ternary constraints, e.g., $\Phi(c) = \oplus_{i,j} [a_i, c_i, b_j]$ (Gokavarapu et al., 24 Nov 2025).
• Syndrome Class: Given a fixed ideal $I$, the syndrome for a word $c$ is $S(c) = \Phi(c) + I$ in the quotient $T/I$. The set of words with a given syndrome $\alpha$ forms a coset $C_\alpha = \{c \in T^n : S(c) = \alpha\}$.

Table: Summary of Coset Partitioning for Representative Frameworks

Context	Syndrome Map	Coset Representative
TGS-codes (Gokavarapu et al., 24 Nov 2025)	$\Phi(c) + I$ in $T/I$	Minimal nonzero element (lattice leader)
Linear codes (Andriatahiny et al., 2018)	$r H^T$	Minimal weight coset leader
Skew-BCH (Gómez-Torrecillas et al., 2016)	Syndromes via skew-eval	Minimal degree right associate
Quotient RM (Gotlib et al., 21 Feb 2025)	Restrictions/lifts	Candidate polynomial lift

Syndrome computation thereby partitions the space in a quotient-algebraic fashion, and the canonical selection of coset leaders underlies the decoding step.

3. Decoding Procedures in Ternary Gamma-Semiring Codes

The quotient-based decoding method for codes over finite commutative TGS proceeds as follows (Gokavarapu et al., 24 Nov 2025):

1. Preprocessing: Enumerate the quotient $T/I$, compute all minimal nonzero $k$-ideals in $L(T)$, and, for each nonzero class in $T/I$, select a unique minimal TGS-weight error vector $e_\alpha$.
2. Syndrome Calculation: Given received $r \in T^n$, compute $\alpha = S(r) = \Phi(r) + I$ in $T/I$.
3. Coset Leader Lookup: If $\alpha = 0 + I$, $r$ is a codeword; otherwise, retrieve $e_\alpha$.
4. Correction: Apply coordinatewise $\oplus$-absorption to remove $e_\alpha$, i.e., $\hat{c} = r \ominus e_\alpha$.
5. Output: $\hat{c}$ is guaranteed to be the codeword closest to $r$ under the minimal TGS-weight.

Correctness is ensured by the structure of the lattice: coset leaders correspond to canonical minimal-weight errors in the distributive lattice, guaranteeing unique decoding in the designed error-correction radius.

4. Extensions and Related Quotient-Based Decoding Methods

Quotient-based decoding methods generalize widely beyond TGS-codes:

• Binary Linear Codes via Gröbner Bases: The code is identified as the zero locus of a binomial ideal $I_C$ in $\mathbb{F}_2[x_1,...,x_n]$. Decoding is accomplished by reducing $X^r - 1$ modulo a Gröbner basis of $I_C$, where the remainder yields the minimal-weight coset leader (Andriatahiny et al., 2018).
• Skew-BCH Convolutional Codes: Decoding proceeds via syndrome computation in the quotient of a skew-polynomial ring. Error locator/evaluator polynomials are computed using the extended Euclidean algorithm in the noncommutative ring, and error correction exploits coset structure defined by the left ideal of the generator polynomial (Gómez-Torrecillas et al., 2016).
• Quotient Reed-Muller Codes: Restriction to subvarieties $\tilde{X} \subset \mathbb{F}_q^n$ produces quotient codes with complicated coset structures, where decoding involves lifting local codewords to ambient codewords, and relies deeply on equidistribution and relative-rank properties of polynomials on varieties (Gotlib et al., 21 Feb 2025).

This diversity demonstrates the foundational role of quotient structures and coset semantics in modern error correction.

5. Computational Complexity and Performance Characteristics

• Precomputation: In TGS and quotient-based codes, constructing the table of coset leaders requires enumeration of the quotient and identification of minimal-weight elements. For TGS-codes, this involves $|T/I|$ coset leader computations (Gokavarapu et al., 24 Nov 2025). For Gröbner basis decoding, the complexity is dominated by the computation of the basis (potentially doubly-exponential, but tractable for sparse or binomial ideals) (Andriatahiny et al., 2018).
• Online Decoding Steps: For TGS-codes, syndrome computation is $O(n^2)$ ternary operations (potentially less for structured maps), coset-leader lookup is $O(1)$ with a table, and error absorption is $O(n)$. In skew-BCH decoding, each Euclidean division is $O(n^2)$ in the base field, with overall complexity $O(nt + t^2)$ for decoding radius $t$ (Gómez-Torrecillas et al., 2016).
• Unique-Decoding Radius: In TGS-codes, $t = \lfloor(\mu(I)-1)/2\rfloor$, with $\mu(I)$ the minimal TGS-weight of a nonzero element in $I$. For linear and Reed-Muller codes, decoding radius is governed by minimum Hamming distance or its analog.

A plausible implication is that quotient-based decoding remains polynomial-time for moderate code parameters and proper algebraic structure, but may become intractable for large dimension or complex quotient structure unless further algebraic or combinatorial properties are exploited.

6. Explicit Example: TGS-Quotient Decoding

Let $T = \{0, a, 1\}$ be a finite TGS with order $0 \oplus a \oplus 1$ and ternary operation $[x,y,z]$ defined so that $[x,y,z] = 0$ if any input is 0, $a$ if exactly one is $a$ (others 1), and $1$ otherwise. The trivial $\Gamma$-action applies; $k$-ideals are $\{0\} \subset I = \{0, a\} \subset T$, so $T/I \cong \{\bar{0}, \bar{1}\}$.

For $n=3$ and code $C = I^3$, minimum distance is 1. The syndrome map is $$\Phi(x_1,x_2,x_3) = [x_1, x_2, a] \oplus [x_2, x_3, a]$$. Received $r=(a,0,a)$ yields syndrome $1+I \neq 0+I$, the coset-leader $e_1 = (0,0,a)$, and correction yields $\hat{c} = (a,0,0) \in C$ (Gokavarapu et al., 24 Nov 2025).

7. Significance and Theoretical Impact

Quotient-based decoding reconceptualizes error correction beyond the scope of classical linear algebraic frameworks, incorporating ternary and higher-arity operations, ideal lattices, and noncommutative ring theory. TGS-codes in particular support parameter sets and decoding behaviors unattainable in finite field codes, group codes, or even standard semiring codes, expanding the landscape for algebraic coding theory. The versatility of quotient frameworks underlies major advances in both code construction and algorithmic decoding, notably in nonlinear, nonbinary, and high-arity codes, as well as algebraic-geometric and convolutional contexts (Gokavarapu et al., 24 Nov 2025, Gotlib et al., 21 Feb 2025, Gómez-Torrecillas et al., 2016, Andriatahiny et al., 2018).