1. Introduction

The famous QR codes, introduced by Prange [1] in 1957, are cyclic BCH codes with code rates greater than or equal to one-half. In addition, the codes generally have large minimum distances so that most of the known QR codes are the best-known codes. In the past decades, several decoding skills have been developed to decode the binary QR codes. The ADAs most used to decode the QR codes are the Newton identities with either Sylvester resultants [3-6,10-11] or Gröbner bases [13], or IFBM algorithm [7-9,12] to determine the error-locator polynomial. The ADA of the (89, 45, 17) QR code [9,12] can correct up to eight errors in GF(2¹¹), because the error-correcting capability of the code is errors, where denotes the greatest integer less than or equal to x, and d = 17 is the minimum Hamming distance of the code. In each decoding procedure, the IFBM algorithm[14] is used to determine the error-locator polynomial of the received sequence[9,12]. Finally, the Chien algorithm[15] is applied to find the roots of the error-locator polynomial. For the QR codes with irreducible generator polynomial, syndrome S₁ = 0 means that the received word has no errors. However, for the (89, 45, 17) QR codes, the syndrome S₁ = 0 means that S₂, S₄, …, are all equal to zero and the zero S₁ does not cause a decoding failure in decoding weight-6 error patterns while using the IFBM algorithm to determine the error-locator polynomial.

The rest parts of the paper are organized as follows: The background of (89, 45, 17) QR codes is briefly given in Section 2. The discussion of the weight-6 error patterns ispresented in Section 3. Finally, this paper concludes with a brief summary in Section 4.

2. Background of the Binary (89, 45, 17) QR Code

The codeword of the binary (n, k, d) QR code is defined algebraically as a multiple of its generator polynomial g(x) with coefficients in GF(2). Let the length of the code n be a prime number of the form n = 8m ± 1, where m is a positive integer and m be the smallest positive integer such that ≡ 1 (mod n). Thus, GF(2^m) is the extension field of GF(2). Also, let k = (n+1)/2 be the message length and d be the minimum Hamming distance or Hamming weight of the code. The generator polynomial as a cyclic code is given by

(1)

where the element β is a primitive nth root of unity in GF(2^m) and Q_n denotes the set of quadratic residues given by Q_n = {jj ≡ x² mod n for 1 ≤ x ≤ (n–1)/2}.

The set Q_n can thus be represented as a disjoint union of cyclotomic cosets, modulo n. These cyclotomic cosets are defined as Q_r = {r2^jj = 0, 1, … , n_r–1}, where n_r is the smallest positive integer such that , n_r divides (n–1)/2, and r is the smallest element in Q_r. The element r is called the representative element of the cyclotomic cosets Q_r. The set S, consisting of all representatives of the QR code, is called the base set of the QR code. These definitions and properties cause the equality relating Q_n to the cyclotomic cosets, modulo n. Let an element α ∈ GF(2¹¹) be a root of the primitive polynomial p(x) = x¹¹ + x² + 1. Then, α generates the multiplicative group of nonzero elements in GF(2¹¹). Also, let an element β = α^u, where u = ( – 1)/n = (2¹¹ – 1)/89 = 23, is a primitive 89th root of unity in GF(2¹¹); that is, β = α²³. The base set of this code is S = {1, 3, 5, 9, 11, 13, 19, 33} and r ∈ S. Therefore, the eight cyclotomic cosets Q_r are shown in Table 1.

Table 1. The cyclotomic cosets of the (89, 45, 17) QR code

Let T = {1, 5, 9, 11} and r ∈ T, the quadratic residue set of the code is given by

(2)

The minimal polynomial g_r(x) can also be expressed as ; therefore, the g(x) consists of four minimum polynomials given below:

(3)

Since the codeword of the QR code is a multiple of the g(x), the codeword polynomial of the QR code of length 89 can be represented by , where C_i ∈ GF(2) for 0 ≤ i ≤ 88, and m(x) = m₄₄x⁴⁴ + + m₁x + m₀ denotes information polynomial, where m_i ∈ GF(2) for 0 ≤ i ≤ 44. In such a representation, this type of codeword is called the non-systematic encoding. In practice, the encoding procedure is often implemented by the use of systematic encoding. Let p(x) = p₄₃x⁴³ + + p₁x + p₀ be the parity-check polynomial, where p_i ∈ GF(2) for 0 ≤ i ≤ 43. Also, let m(x)x^n-k divide by g(x), then we get the following identity:

(4)

Multiplying both sides of (4) by x^k and using xⁿ = 1, Then, it yields d(x)x^k + m(x) = (q(x)x^k)g(x). The term d(x)x^k + m(x), which is a multiple of g(x), has m(x) in its lower k bits and p(x) = d(x)x^k in its higher n – k bits. Thus, the codeword can be represented by the equation below.

(5)

As shown in (5), this form of the codeword is called systematic encoding. Now, let a codeword be transmitted through an additive white Gaussian noise (AWGN) channel, and obtain a received word of the form r(x) = c(x) + e(x), where e(x) = e₈₈x⁸⁸ ++ e₁x + e₀ is the error polynomial and e_i ∈ GF(2). For simplification, the polynomial form can be expressed as the vector form. For example, c(x) can be expressed as c = (c₈₈, …, c₁, c₀). The syndromes or known syndromes of the code are defined by

(6)

where i (mod 89) ∈ Q₈₉. If i ∉ Q₈₉, the syndromes are called unknown syndromes. All the known and unknown syndromes can be expressed as some powers of S₁, S₅, S₉, S₁₁, and S₃, S₁₃, S₁₉, S₃₃, called the primary known and unknown syndromes, respectively. Notice that S₀ = 0 or 1 depends on the fact that v is even or odd, where v ≤ t is the actual number of errors to be corrected and 1 ≤ v ≤ 8.

If there are v errors in r(x), then e(x) has v nonzero terms over GF(2); that is, , where 0 ≤ r₁ < r₂ < ... < r_v ≤ n–1. For i ∈ Q_n, the syndrome can be written as . Assuming that v errors occur, the error-locator polynomial L_v(x) is defined by

(7)

where are called the error locators, the σ_j are called the elementary symmetric functions for 1 ≤ j ≤ v, and σ₀ = 1. The roots of L_v(x) are the inverse of the v error locators {X_j}.

To determine the error-locator polynomial, the steps of the IFBM algorithm [14] are summarized as below:

1. Initialize k = 0, μ⁽⁰⁾(x) = 1, λ⁽⁰⁾(x)=1, l⁽⁰⁾ = 0, γ⁽⁰⁾ = 1.

2. Set k = k + 1. If k > 2t, then go to stop. Otherwise Compute

(8)

where μ_j^(k–1) is the coefficient of μ^(k–1)(x).

3. Compute

(9)

4. Compute

(10)

5. Set k = k + 1. If k ≤ 2t, then go to 2. Otherwise stop.

3. Discussion of the Weight-6 Error Patterns

The ADA given in [9,12] utilized the IFBM algorithm to determine the error-locator polynomial. In order to apply the IFBM algorithm, the consecutive syndromes S_i for 1 ≤ i ≤ 16 need to be computed. Among them, the unknown syndromes S₃, S₆, S₇, S₁₂, S₁₃, S₁₄, and S₁₅ can be expressed as some powers of the primary unknown syndrome S₃ or S₁₃ that cannot be computed directly from the received word. The determination of the S₃ and S₁₃ is given in [9,12].

For the (89, 45, 17) QR code, a C++ program shows that over 200,000 weight-6 error patterns will cause a zero S₁. However, the zero S₁ does not cause a decoding failure while using the IFBM algorithm to determine the error-locator polynomial. The following example demonstrates the fact.

Example 1

If a message m(x) = 10, by (4) and (5), then one obtains the systematic codeword c(x) = x⁸⁶ + x⁸⁵ + x⁸⁴ + x⁸¹ + x⁷⁷ + x⁷⁵ + x⁷² + x⁶⁵ + x⁶² + x⁶⁰ + x⁵⁶ + x⁵³ + x⁵² + x⁵¹ + x⁴⁷ + x⁴⁵ + x³ + x. If there is a weight-6 error pattern e(x) = x²⁶ + x¹⁵ + x³ + x² + x + 1 occurred in the received word, then one obtains r(x) = x⁸⁶ + x⁸⁵ + x⁸⁴ + x⁸¹ + x⁷⁷ + x⁷⁵ + x⁷² + x⁶⁵ + x⁶² + x⁶⁰ + x⁵⁶ + x⁵³ + x⁵² + x⁵¹ + x⁴⁷ + x⁴⁵ + x²⁶ + x¹⁵ + x² + 1.

By (6), one obtains S₁ = S₂ = S₄ = S₈ = 0, S₃ = 57, S₅ = 746, S₆ = S₃² = 1345, S₇ = S₃³² = 1938, S₉ = 711, S₁₀ = S₅² = 1766, S₁₁ = 376, S₁₂ = S₃⁴ = 686, where the values of all syndromes are expressed in decimal number. Then, by using the IFBM algorithm to obtain the error-locator polynomial L₆(x), one has the following detailed steps.

1. Define the initial values as follows: k = 0, μ⁽⁰⁾(x) = 1, λ⁽⁰⁾(x)=1, l⁽⁰⁾ = 0, and γ⁽⁰⁾ = 1.

2. Set k = 1. By (8), one has δ⁽¹⁾ = 0.

3. By (9), compute μ⁽¹⁾(x) = γ⁽⁰⁾μ⁽⁰⁾(x) – δ⁽¹⁾λ⁽⁰⁾(x)x = γ⁽⁰⁾μ⁽⁰⁾(x) = 1∙1 = 1.

4. By (10), δ⁽¹⁾ = 0. Then, compute λ⁽¹⁾(x) = xλ⁽⁰⁾(x) = x1 = x, l⁽¹⁾ = l⁽⁰⁾ = 0, and γ⁽¹⁾ = γ⁽⁰⁾ = 1, respectively. Go to step 2.

2. Set k = 2. By (8), one has δ⁽²⁾ = 0.

3. By (9), compute μ⁽²⁾(x) = γ⁽¹⁾μ⁽¹⁾(x) – δ⁽²⁾λ⁽¹⁾(x)x = γ⁽¹⁾μ⁽¹⁾(x) = 1∙1 = 1.

4. By (10), δ⁽²⁾ = 0. Then, compute λ⁽²⁾(x) = xλ⁽¹⁾(x) =x², l⁽²⁾ = l⁽¹⁾ = 0, and γ⁽²⁾ = γ⁽¹⁾ = 1, respectively. Go to step 2.

2. Set k = 3. By (8), one has δ⁽²⁾ = S₃.

3. By (9), compute μ⁽³⁾(x) = γ⁽²⁾μ⁽²⁾(x) – δ⁽³⁾λ⁽²⁾(x)x = 1∙1 – S₃x²x = 1 + S₃x³. Notice that the addition and subtraction are the same in GF(2).

4. By (10), δ⁽³⁾ = S₃ ≠ 0 and 2l⁽²⁾ = 0 ≤ k – 1 = 3 – 1 = 2. Then, compute λ⁽³⁾(x) = μ^(k-1)(x) = μ⁽²⁾(x) = 1, l⁽³⁾ = k – l⁽²⁾ = 3 – 0 = 3, and γ⁽³⁾ = δ⁽³⁾ = S₃, respectively. Go to step 2.

2. Set k = 4. By (8), one has δ⁽⁴⁾ = 0.

3. By (9), compute μ⁽⁴⁾(x) = γ⁽³⁾μ⁽³⁾(x) – δ⁽⁴⁾λ⁽³⁾(x)x = γ⁽³⁾μ⁽³⁾(x) = S₃(1 + S₃x³) = S₃ + S₃²x³.

4. By (10), δ⁽⁴⁾ = 0. Then, compute λ⁽⁴⁾(x) = xλ⁽³⁾(x) = x, l⁽⁴⁾ = l⁽³⁾ = 3, and γ⁽⁴⁾ = γ⁽³⁾ = S₃, respectively. Go to step 2.

2. Set k = 5. By (8), one has δ⁽⁵⁾ = S₃S₅.

3. By (9), compute μ⁽⁵⁾(x) = γ⁽⁴⁾μ⁽⁴⁾(x) – δ⁽⁵⁾λ⁽⁴⁾(x)x = S₃(S₃ +S₃²x³) – S₃S₅xx = S₃² + S₃S₅x² + S₃³x³.

4. By (10), 2l⁽⁴⁾ = 6 > k – 1 = 5 – 1 = 4. Then, compute λ⁽⁵⁾(x) = xλ⁽⁴⁾(x) = xx = x², l⁽⁵⁾ = l⁽⁴⁾ = 3, and γ⁽⁵⁾ = γ⁽⁴⁾ = S₃, respectively. Go to step 2.

2. Set k = 6. By (8) and S₆ = S₃², one has δ⁽⁶⁾ = 0.

3. By (9), compute μ⁽⁶⁾(x) = γ⁽⁵⁾μ⁽⁵⁾(x) – δ⁽⁶⁾λ⁽⁵⁾(x)x = γ⁽⁵⁾μ⁽⁵⁾(x) = S₃(S₃² + S₃S₅x² + S₃³x³) = S₃³ + S₃²S₅x² + S₃⁴x³.

4. By (10), δ⁽⁶⁾ = 0. Then, compute λ⁽⁶⁾(x) = xλ⁽⁵⁾(x) = x³, l⁽⁶⁾ = l⁽⁵⁾ = 3, and γ⁽⁶⁾ = γ⁽⁵⁾, respectively. Go to step 2.

2. Set k = 7. By (8) and S₇ = S₃³², one has δ⁽⁷⁾ = S₃³⁵ + S₃²S₅².

3. By (9), compute μ⁽⁷⁾(x) = γ⁽⁶⁾μ⁽⁶⁾(x) – δ⁽⁷⁾λ⁽⁶⁾(x)x = S₃(S₃³ + S₃²S₅x² + S₃⁴x³) – (S₃³⁵ + S₃²S₅²)x³x = S₃⁴ + S₃³S₅x² + S₃⁵x³ + (S₃³⁵ + S₃²S₅²)x⁴.

4. By (10), δ⁽⁷⁾ ≠ 0 and 2l⁽⁶⁾ = 6 ≤ k – 1 = 7 – 1 = 6. Then, compute λ⁽⁷⁾(x) = μ⁽⁶⁾(x) = S₃³ + S₃²S₅x² + S₃⁴x³, l⁽⁷⁾ = k – l⁽⁶⁾ = 7 – 3 = 4, and γ⁽⁷⁾ = δ⁽⁷⁾, respectively. Go to step 2.

2. Set k = 8. By (8), one has δ⁽⁸⁾ = 0.

3. By (9), compute μ⁽⁸⁾(x) = γ⁽⁷⁾μ⁽⁷⁾(x) – δ⁽⁸⁾λ⁽⁷⁾(x)x = γ⁽⁷⁾μ⁽⁷⁾(x) = (S₃³⁵ + S₃²S₅²)(S₃⁴ + S₃³S₅x² + S₃⁵x³ + (S₃³⁵ + S₃²S₅²)x⁴) = (S₃³⁹ + S₃⁶S₅²) + (S₃³⁸S₅ + S₃⁵S₅³)x² + (S₃⁴⁰ + S₃⁷S₅²)x³ + (S₃⁷⁰ + S₃⁴S₅⁴)x⁴.

4. By (10), δ⁽⁸⁾ = 0. Then, compute λ⁽⁸⁾(x) = xλ⁽⁷⁾(x) = x(S₃³ + S₃²S₅x² + S₃⁴x³) = S₃³x + S₃²S₅x³ + S₃⁴x⁴, l⁽⁸⁾ = l⁽⁷⁾ = 4, and γ⁽⁸⁾ = γ⁽⁷⁾ = S₃³⁵ + S₃²S₅², respectively.

2. Set k = 9. By (8), one has δ⁽⁹⁾ = S₃³⁹S₉ + S₃⁶S₅²S₉ + S₃³⁷S₅³ +S₃⁴² + S₃⁹S₅² + S₃⁴S₅⁵.

3. By (9), compute μ⁽⁹⁾(x) = γ⁽⁸⁾μ⁽⁸⁾(x) – δ⁽⁹⁾λ⁽⁸⁾(x)x = (S₃³⁵ + S₃²S₅²)((S₃³⁹ + S₃⁶S₅²) + (S₃³⁸S₅ + S₃⁵S₅³)x² – (S₃⁴⁰ + S₃⁷S₅²)x³ + (S₃⁷⁰ + S₃⁴S₅⁴)x⁴) – (S₃³⁹S₉ + S₃⁶S₅²S₉ + S₃³⁷S₅³ +S₃⁴² + S₃⁹S₅² + S₃⁴S₅⁵)(S₃³x + S₃²S₅x³ + S₃⁴x⁴)x = (S₃⁷⁴ + S₃⁸S₅⁴) + (S₃⁷³S₅ + S₃⁴²S₉ + S₃⁹S₅²S₉ + S₃⁴⁰S₅³ + S₃⁴⁵ + S₃¹²S₅²)x² + (S₃⁷⁵ + S₃⁹S₅⁴)x³ + (S₃¹⁰⁵ + S₃⁷²S₅² + S₃⁴¹S₅S₉ + S₃⁸S₅³S₉ + S₃⁴⁴S₅ + S₃¹¹S₅³)x⁴ + (S₃⁴³S₉ + S₃¹⁰S₅²S₉ + S₃⁴¹S₅³ + S₃⁴⁶ + S₃¹³S₅² + S₃⁸S₅⁵)x⁵.

4. By (10), δ⁽⁹⁾ ≠ 0 and 2l⁽⁸⁾ = 8 ≤ k – 1 = 9 – 1 = 8. Then, compute λ⁽⁹⁾(x) =μ⁽⁸⁾(x) = (S₃³⁹ + S₃⁶S₅²) + (S₃³⁸S₅ + S₃⁵S₅³)x² + (S₃⁴⁰ + S₃⁷S₅²)x³ + (S₃⁷⁰ + S₃⁴S₅⁴)x⁴, l⁽⁹⁾ = k – l⁽⁸⁾ = 9 – 4 = 5, and γ⁽⁹⁾ = δ⁽⁹⁾, respectively. Go to step 2.

2. Set k = 10. By (8) and S₁₀ = S₅², one has δ⁽¹⁰⁾ = 0.

3. By (9), compute μ⁽¹⁰⁾(x) = γ⁽⁹⁾μ⁽⁹⁾(x) – δ⁽¹⁰⁾λ⁽⁹⁾(x)x = γ⁽⁹⁾μ⁽⁹⁾(x) = (S₃³⁹S₉ + S₃⁶S₅²S₉ + S₃³⁷S₅³ +S₃⁴² + S₃⁹S₅² + S₃⁴S₅⁵)((S₃⁷⁴ + S₃⁸S₅⁴) + (S₃⁷³S₅ + S₃⁴²S₉ + S₃⁹S₅²S₉ + S₃⁴⁰S₅³ + S₃⁴⁵ + S₃¹²S₅²)x² + (S₃⁷⁵ + S₃⁹S₅⁴)x³ + (S₃¹⁰⁵ + S₃⁷²S₅² + S₃⁴¹S₅S₉ + S₃⁸S₅³S₉ + S₃⁴⁴S₅ + S₃¹¹S₅³)x⁴ + (S₃⁴³S₉ + S₃¹⁰S₅²S₉ + S₃⁴¹S₅³ + S₃⁴⁶ + S₃¹³S₅² + S₃⁸S₅⁵)x⁵) = (S₃¹¹³S₉ + S₃¹¹⁶ + S₃¹²S₅⁹ + S₃¹⁷S₅⁶ + S₃⁴⁵S₅⁷ + S₃⁵⁰S₅⁴ + S₃⁷⁸S₅⁵ + S₃⁸³S₅² + S₃¹¹¹S₅³ + S₃¹⁴S₅⁶S₉ + S₃⁴⁷S₅⁴S₉ + S₃⁸⁰S₅²S₉) + (S₃¹¹⁵S₅ + S₃¹¹²S₅S₉ + S₃¹¹⁰S₅⁴ + S₃⁸⁷ + S₃⁸²S₅³ + S₃⁸¹S₉² + S₃⁷⁹S₅³S₉ + S₃⁴⁹S₅⁵ + S₃⁴⁶S₅⁵S₉ + S₃⁴⁴S₅⁸ + S₃²¹S₅⁴ + S₃¹⁶S₅⁷ + S₃¹⁵S₅⁴S₉² + S₃¹³S₅⁷S₉)x² + (S₃¹¹⁷ + S₉S₃¹¹⁴ + S₃¹¹²S₅³ + S₃⁸⁴S₅² + S₉S₃⁸¹S₅² + S₃⁷⁹S₅⁵ + S₃⁵¹S₅⁴ + S₉S₃⁴⁸S₅⁴ + S₃⁴⁶S₅⁷ + S₃¹⁸S₅⁶ + S₉S₃¹⁵S₅⁶ + S₃¹³S₅⁹)x³ + (S₃¹⁴⁷ + S₃¹⁴⁴S₉ + S₃¹⁴²S₅³ + S₃⁸⁶S₅ + S₃⁸⁰S₅S₉² + S₃⁷⁶S₅⁷ + S₃²⁰S₅⁵ + S₃¹⁵S₅⁸ + S₃¹⁴S₅⁵S₉² + S₃¹²S₅⁸S₉)x⁴ + (S₃⁸⁸ + S₃⁸⁵S₉ + S₃⁸⁰S₅³S₉ + S₃⁷⁸S₅⁶ + S₃⁵²S₅²S₉ + S₃⁴⁹S₅²S₉² + S₃⁴⁷S₅⁵S₉ + S₃⁴⁵S₉ + S₃⁴²S₉² + S₃⁴⁰S₅³S₉ + S₃²²S₅⁴ + S₃¹⁶S₅⁴S₉² + S₃¹²S₅¹⁰ + S₃¹²S₅²S₉ + S₃⁹S₅²S₉² + S₃⁷S₅⁵S₉)x⁵.

4. By (10), δ⁽¹⁰⁾ = 0. Then, compute λ⁽¹⁰⁾(x) = xλ⁽⁹⁾(x) = (S₃³⁹ + S₃⁶S₅²)x + (S₃³⁸S₅ + S₃⁵S₅³)x³ + (S₃⁴⁰ + S₃⁷S₅²)x⁴ + (S₃⁷⁰ + S₃⁴S₅⁴)x⁵, l⁽¹⁰⁾ = l⁽⁹⁾ = 5, and γ⁽¹⁰⁾ = γ⁽⁹⁾, respectively.

2. Set k = 11. By (8), one has δ⁽¹¹⁾ = S₃¹⁷⁹ + S₃¹⁷⁶S₉ + S₃¹⁷⁴S₅³ + S₃¹¹⁸S₅ + S₁₁S₃¹¹⁶ + S₃¹¹⁵S₅S₉ + S₁₁S₃¹¹³S₉ + S₁₁S₃¹¹¹S₅³ + S₃¹¹⁰S₅⁴S₉ + S₃¹⁰⁸S₅⁷ + S₃⁹⁰ + S₃⁸⁷S₉ + S₃⁸⁴S₉² + S₁₁S₃⁸³S₅² + S₃⁸²S₅³S₉ + S₃⁸¹S₉³ + S₃⁸⁰S₅⁶ + S₁₁S₃⁸⁰S₅²S₉ + S₃⁷⁹S₅³S₉² + S₁₁S₃⁷⁸S₅⁵ + S₃⁵²S₅⁵ + S₁₁S₃⁵⁰S₅⁴ + S₃⁴⁹S₅⁵S₉ + S₃⁴⁷S₅⁸ + S₁₁S₃⁴⁷S₅⁴S₉ + S₁₁S₃⁴⁵S₅⁷ + S₃²⁴S₅⁴ + S₃²¹S₅⁴S₉ + S₃¹⁸S₅⁴S₉² + S₁₁S₃¹⁷S₅⁶ + S₃¹⁶S₅⁷S₉ + S₃¹⁵S₅⁴S₉³ + S₃¹⁴S₅¹⁰ + S₁₁S₃¹⁴S₅⁶S₉ + S₃¹³S₅⁷S₉² + S₁₁S₃¹²S₅⁹.

3. μ⁽¹¹⁾(x) = γ⁽¹⁰⁾μ⁽¹⁰⁾(x) – δ⁽¹¹⁾λ⁽¹⁰⁾(x)x = (S₃³⁹S₉ + S₃⁶S₅²S₉ + S₃³⁷S₅³ +S₃⁴² + S₃⁹S₅² + S₃⁴S₅⁵)((S₃¹¹³S₉ + S₃¹¹⁶ + S₃¹²S₅⁹ + S₃¹⁷S₅⁶ + S₃⁴⁵S₅⁷ + S₃⁵⁰S₅⁴ + S₃⁷⁸S₅⁵ + S₃⁸³S₅² + S₃¹¹¹S₅³ + S₃¹⁴S₅⁶S₉ + S₃⁴⁷S₅⁴S₉ + S₃⁸⁰S₅²S₉) + (S₃¹¹⁵S₅ + S₃¹¹²S₅S₉ + S₃¹¹⁰S₅⁴ + S₃⁸⁷ + S₃⁸²S₅³ + S₃⁸¹S₉² + S₃⁷⁹S₅³S₉ + S₃⁴⁹S₅⁵ + S₃⁴⁶S₅⁵S₉ + S₃⁴⁴S₅⁸ + S₃²¹S₅⁴ + S₃¹⁶S₅⁷ + S₃¹⁵S₅⁴S₉² + S₃¹³S₅⁷S₉)x² + (S₃¹¹⁷ + S₉S₃¹¹⁴ + S₃¹¹²S₅³ + S₃⁸⁴S₅² + S₉S₃⁸¹S₅² + S₃⁷⁹S₅⁵ + S₃⁵¹S₅⁴ + S₉S₃⁴⁸S₅⁴ + S₃⁴⁶S₅⁷ + S₃¹⁸S₅⁶ + S₉S₃¹⁵S₅⁶ + S₃¹³S₅⁹)x³ + (S₃¹⁴⁷ + S₃¹⁴⁴S₉ + S₃¹⁴²S₅³ + S₃⁸⁶S₅ + S₃⁸⁰S₅S₉² + S₃⁷⁶S₅⁷ + S₃²⁰S₅⁵ + S₃¹⁵S₅⁸ + S₃¹⁴S₅⁵S₉² + S₃¹²S₅⁸S₉)x⁴ + (S₃⁸⁸ + S₃⁸²S₉² + S₃⁷⁸S₅⁶ + S₃²²S₅⁴ + S₃¹⁶S₅⁴S₉² + S₃¹²S₅¹⁰)x⁵) – (S₃¹⁷⁹ + S₃¹⁷⁶S₉ + S₃¹⁷⁴S₅³ + S₃¹¹⁸S₅ + S₁₁S₃¹¹⁶ + S₃¹¹⁵S₅S₉ + S₁₁S₃¹¹³S₉ + S₁₁S₃¹¹¹S₅³ + S₃¹¹⁰S₅⁴S₉ + S₃¹⁰⁸S₅⁷ + S₃⁹⁰ + S₃⁸⁷S₉ + S₃⁸⁴S₉² + S₁₁S₃⁸³S₅² + S₃⁸²S₅³S₉ + S₃⁸¹S₉³ + S₃⁸⁰S₅⁶ + S₁₁S₃⁸⁰S₅²S₉ + S₃⁷⁹S₅³S₉² + S₁₁S₃⁷⁸S₅⁵ + S₃⁵²S₅⁵ + S₁₁S₃⁵⁰S₅⁴ + S₃⁴⁹S₅⁵S₉ + S₃⁴⁷S₅⁸ + S₁₁S₃⁴⁷S₅⁴S₉ + S₁₁S₃⁴⁵S₅⁷ + S₃²⁴S₅⁴ + S₃²¹S₅⁴S₉ + S₃¹⁸S₅⁴S₉² + S₁₁S₃¹⁷S₅⁶ + S₃¹⁶S₅⁷S₉ + S₃¹⁵S₅⁴S₉³ + S₃¹⁴S₅¹⁰ + S₁₁S₃¹⁴S₅⁶S₉ + S₃¹³S₅⁷S₉² + S₁₁S₃¹²S₅⁹)((S₃³⁹ + S₃⁶S₅²)x + (S₃³⁸S₅ + S₃⁵S₅³)x³ + (S₃⁴⁰ + S₃⁷S₅²)x⁴ + (S₃⁷⁰ + S₃⁴S₅⁴)x⁵)x = (S₃¹⁵⁸ + S₃¹⁶S₅¹⁴ + S₃²⁶S₅⁸ + S₃¹⁴⁸S₅⁶ + S₃¹⁵²S₉² + S₃²⁰S₅⁸S₉²) + (S₃²¹⁸ + S₃²¹⁵S₉ + S₃²¹³S₅³ + S₃¹⁸⁵S₅² + S₃¹⁸²S₅²S₉ + S₃¹⁸⁰S₅⁵ + S₁₁S₃¹⁵⁵ + S₃¹⁵⁴S₅S₉ + S₁₁S₃¹⁵²S₉ + S₃¹⁵¹S₅S₉² + S₁₁S₃¹⁵⁰S₅³ + S₃¹⁴⁹S₅⁴S₉ + S₃⁸⁶S₅⁸ + S₃⁸³S₅⁸S₉ + S₃⁸¹S₅¹¹ + S₃⁵³S₅¹⁰ + S₃⁵⁰S₅¹⁰S₉ + S₃⁴⁸S₅¹³ + S₁₁S₃²³S₅⁸ + S₃²²S₅⁹S₉ + S₁₁S₃²⁰S₅⁸S₉ + S₃¹⁹S₅⁹S₉² + S₁₁S₃¹⁸S₅¹¹ + S₃¹⁷S₅¹²S₉)x² + (S₃¹⁵⁹ + S₃¹⁵³S₉² + S₃¹⁴⁹S₅⁶ + S₃²⁷S₅⁸ + S₃²¹S₅⁸S₉² + S₃¹⁷S₅¹⁴)x³ + (S₃²¹⁷S₅ + S₃²¹⁴S₅S₉ + S₃²¹²S₅⁴ + S₃¹⁸⁹ + S₃¹⁸⁴S₅³ + S₃¹⁸³S₉² + S₃¹⁸¹S₅³S₉ + S₁₁S₃¹⁵⁴S₅ + S₃¹⁵³S₅²S₉ + S₁₁S₃¹⁵¹S₅S₉ + S₃¹⁵⁰S₅²S₉² + S₁₁S₃¹⁴⁹S₅⁴ + S₃¹⁴⁸S₅⁵S₉ + S₃⁸⁵S₅⁹ + S₃⁸²S₅⁹S₉ + S₃⁸⁰S₅¹² + S₃⁵⁷S₅⁸ + S₃⁵²S₅¹¹ + S₃⁵¹S₅⁸S₉² + S₃⁴⁹S₅¹¹S₉ + S₁₁S₃²²S₅⁹ + S₃²¹S₅¹⁰S₉ + S₁₁S₃¹⁹S₅⁹S₉ + S₃¹⁸S₅¹⁰S₉² + S₁₁S₃¹⁷S₅¹² + S₃¹⁶S₅¹³S₉)x⁴ + (S₃²¹⁹ + S₉S₃²¹⁶ + S₃²¹⁴S₅³ + S₃¹⁸⁶S₅² + S₉S₃¹⁸³S₅² + S₃¹⁸¹S₅⁵ + S₃¹⁵⁸S₅ + S₁₁S₃¹⁵⁶ + S₉S₃¹⁵⁵S₅ + S₁₁S₉S₃¹⁵³ + S₁₁S₃¹⁵¹S₅³ + S₉S₃¹⁵⁰S₅⁴ + S₃¹⁴⁸S₅⁷ + S₃⁸⁷S₅⁸ + S₉S₃⁸⁴S₅⁸ + S₃⁸²S₅¹¹ + S₃⁵⁴S₅¹⁰ + S₉S₃⁵¹S₅¹⁰ + S₃⁴⁹S₅¹³ + S₃²⁶S₅⁹ + S₁₁S₃²⁴S₅⁸ + S₉S₃²³S₅⁹ + S₁₁S₉S₃²¹S₅⁸ + S₁₁S₃¹⁹S₅¹¹ + S₉S₃¹⁸S₅¹² + S₃¹⁶S₅¹⁵)x⁵ + (S₃²⁴⁹ + S₃²⁴⁶S₉ + S₃²⁴⁴S₅³ + S₃¹⁸⁸S₅ + S₁₁S₃¹⁸⁶ + S₃¹⁸⁵S₅S₉ + S₃¹⁸³S₅⁴ + S₁₁S₃¹⁸³S₉ + S₁₁S₃¹⁸¹S₅³ + S₃¹⁶⁰ + S₃¹⁵⁷S₉ + S₃¹⁵⁴S₉² + S₁₁S₃¹⁵³S₅² + S₃¹⁵²S₅³S₉ + S₃¹⁵¹S₉³ + S₃¹⁵⁰S₅⁶ + S₁₁S₃¹⁵⁰S₅²S₉ + S₃¹⁴⁹S₅³S₉² + S₁₁S₃¹⁴⁸S₅⁵ + S₃¹¹⁷S₅⁸ + S₃¹¹⁴S₅⁸S₉ + S₃¹¹²S₅¹¹ + S₃⁵⁶S₅⁹ + S₁₁S₃⁵⁴S₅⁸ + S₃⁵³S₅⁹S₉ + S₃⁵¹S₅¹² + S₁₁S₃⁵¹S₅⁸S₉ + S₁₁S₃⁴⁹S₅¹¹ + S₃²⁸S₅⁸ + S₃²⁵S₅⁸S₉ + S₃²²S₅⁸S₉² + S₁₁S₃²¹S₅¹⁰ + S₃²⁰S₅¹¹S₉ + S₃¹⁹S₅⁸S₉³ + S₃¹⁸S₅¹⁴ + S₁₁S₃¹⁸S₅¹⁰S₉ + S₃¹⁷S₅¹¹S₉² + S₁₁S₃¹⁶S₅¹³)x⁶.

4. By (10), δ⁽¹¹⁾ ≠ 0 and 2l⁽¹⁰⁾ = 10 ≤ k – 1 = 11 – 1 = 10. Then, compute λ⁽¹¹⁾(x) = μ⁽¹⁰⁾(x), l⁽¹¹⁾ = k – l⁽¹⁰⁾ = 11 – 5 = 6, and γ⁽¹¹⁾ = δ⁽¹¹⁾, respectively. Go to step 2.

2. Set k = 12. One has δ⁽¹²⁾ = 0.

3. By (15) in[14], compute the error-locator polynomial of degree 6, μ⁽¹²⁾(x) = γ⁽¹¹⁾μ⁽¹¹⁾(x) – δ⁽¹²⁾λ⁽¹¹⁾(x)x = γ⁽¹¹⁾μ⁽¹¹⁾(x) = (S₃¹⁷⁹ + S₃¹⁷⁶S₉ + S₃¹⁷⁴S₅³ + S₃¹¹⁸S₅ + S₁₁S₃¹¹⁶ + S₃¹¹⁵S₅S₉ + S₁₁S₃¹¹³S₉ + S₁₁S₃¹¹¹S₅³ + S₃¹¹⁰S₅⁴S₉ + S₃¹⁰⁸S₅⁷ + S₃⁹⁰ + S₃⁸⁷S₉ + S₃⁸⁴S₉² + S₁₁S₃⁸³S₅² + S₃⁸²S₅³S₉ + S₃⁸¹S₉³ + S₃⁸⁰S₅⁶ + S₁₁S₃⁸⁰S₅²S₉ + S₃⁷⁹S₅³S₉² + S₁₁S₃⁷⁸S₅⁵ + S₃⁵²S₅⁵ + S₁₁S₃⁵⁰S₅⁴ + S₃⁴⁹S₅⁵S₉ + S₃⁴⁷S₅⁸ + S₁₁S₃⁴⁷S₅⁴S₉ + S₁₁S₃⁴⁵S₅⁷ + S₃²⁴S₅⁴ + S₃²¹S₅⁴S₉ + S₃¹⁸S₅⁴S₉² + S₁₁S₃¹⁷S₅⁶ + S₃¹⁶S₅⁷S₉ + S₃¹⁵S₅⁴S₉³ + S₃¹⁴S₅¹⁰ + S₁₁S₃¹⁴S₅⁶S₉ + S₃¹³S₅⁷S₉² + S₁₁S₃¹²S₅⁹)((S₃¹⁵⁸ + S₃¹⁶S₅¹⁴ + S₃²⁶S₅⁸ + S₃¹⁴⁸S₅⁶ + S₃¹⁵²S₉² + S₃²⁰S₅⁸S₉²) + (S₃²¹⁸ + S₃²¹⁵S₉ + S₃²¹³S₅³ + S₃¹⁸⁵S₅² + S₃¹⁸²S₅²S₉ + S₃¹⁸⁰S₅⁵ + S₁₁S₃¹⁵⁵ + S₃¹⁵⁴S₅S₉ + S₁₁S₃¹⁵²S₉ + S₃¹⁵¹S₅S₉² + S₁₁S₃¹⁵⁰S₅³ + S₃¹⁴⁹S₅⁴S₉ + S₃⁸⁶S₅⁸ + S₃⁸³S₅⁸S₉ + S₃⁸¹S₅¹¹ + S₃⁵³S₅¹⁰ + S₃⁵⁰S₅¹⁰S₉ + S₃⁴⁸S₅¹³ + S₁₁S₃²³S₅⁸ + S₃²²S₅⁹S₉ + S₁₁S₃²⁰S₅⁸S₉ + S₃¹⁹S₅⁹S₉² + S₁₁S₃¹⁸S₅¹¹ + S₃¹⁷S₅¹²S₉)x² + (S₃¹⁵⁹ + S₃¹⁵³S₉² + S₃¹⁴⁹S₅⁶ + S₃²⁷S₅⁸ + S₃²¹S₅⁸S₉² + S₃¹⁷S₅¹⁴)x³ + (S₃²¹⁷S₅ + S₃²¹⁴S₅S₉ + S₃²¹²S₅⁴ + S₃¹⁸⁹ + S₃¹⁸⁴S₅³ + S₃¹⁸³S₉² + S₃¹⁸¹S₅³S₉ + S₁₁S₃¹⁵⁴S₅ + S₃¹⁵³S₅²S₉ + S₁₁S₃¹⁵¹S₅S₉ + S₃¹⁵⁰S₅²S₉² + S₁₁S₃¹⁴⁹S₅⁴ + S₃¹⁴⁸S₅⁵S₉ + S₃⁸⁵S₅⁹ + S₃⁸²S₅⁹S₉ + S₃⁸⁰S₅¹² + S₃⁵⁷S₅⁸ + S₃⁵²S₅¹¹ + S₃⁵¹S₅⁸S₉² + S₃⁴⁹S₅¹¹S₉ + S₁₁S₃²²S₅⁹ + S₃²¹S₅¹⁰S₉ + S₁₁S₃¹⁹S₅⁹S₉ + S₃¹⁸S₅¹⁰S₉² + S₁₁S₃¹⁷S₅¹² + S₃¹⁶S₅¹³S₉)x⁴ + (S₃²¹⁹ + S₉S₃²¹⁶ + S₃²¹⁴S₅³ + S₃¹⁸⁶S₅² + S₉S₃¹⁸³S₅² + S₃¹⁸¹S₅⁵ + S₃¹⁵⁸S₅ + S₁₁S₃¹⁵⁶ + S₉S₃¹⁵⁵S₅ + S₁₁S₉S₃¹⁵³ + S₁₁S₃¹⁵¹S₅³ + S₉S₃¹⁵⁰S₅⁴ + S₃¹⁴⁸S₅⁷ + S₃⁸⁷S₅⁸ + S₉S₃⁸⁴S₅⁸ + S₃⁸²S₅¹¹ + S₃⁵⁴S₅¹⁰ + S₉S₃⁵¹S₅¹⁰ + S₃⁴⁹S₅¹³ + S₃²⁶S₅⁹ + S₁₁S₃²⁴S₅⁸ + S₉S₃²³S₅⁹ + S₁₁S₉S₃²¹S₅⁸ + S₁₁S₃¹⁹S₅¹¹ + S₉S₃¹⁸S₅¹² + S₃¹⁶S₅¹⁵)x⁵ + (S₃²⁴⁹ + S₃²⁴⁶S₉ + S₃²⁴⁴S₅³ + S₃¹⁸⁸S₅ + S₁₁S₃¹⁸⁶ + S₃¹⁸⁵S₅S₉ + S₃¹⁸³S₅⁴ + S₁₁S₃¹⁸³S₉ + S₁₁S₃¹⁸¹S₅³ + S₃¹⁶⁰ + S₃¹⁵⁷S₉ + S₃¹⁵⁴S₉² + S₁₁S₃¹⁵³S₅² + S₃¹⁵²S₅³S₉ + S₃¹⁵¹S₉³ + S₃¹⁵⁰S₅⁶ + S₁₁S₃¹⁵⁰S₅²S₉ + S₃¹⁴⁹S₅³S₉² + S₁₁S₃¹⁴⁸S₅⁵ + S₃¹¹⁷S₅⁸ + S₃¹¹⁴S₅⁸S₉ + S₃¹¹²S₅¹¹ + S₃⁵⁶S₅⁹ + S₁₁S₃⁵⁴S₅⁸ + S₃⁵³S₅⁹S₉ + S₃⁵¹S₅¹² + S₁₁S₃⁵¹S₅⁸S₉ + S₁₁S₃⁴⁹S₅¹¹ + S₃²⁸S₅⁸ + S₃²⁵S₅⁸S₉ + S₃²²S₅⁸S₉² + S₁₁S₃²¹S₅¹⁰ + S₃²⁰S₅¹¹S₉ + S₃¹⁹S₅⁸S₉³ + S₃¹⁸S₅¹⁴ + S₁₁S₃¹⁸S₅¹⁰S₉ + S₃¹⁷S₅¹¹S₉² + S₁₁S₃¹⁶S₅¹³)x⁶) = S₁₁S₃²⁷⁴ + S₃²⁷⁶S₅ + S₃²⁴⁵S₉ + S₃³³⁴S₉ + S₃²⁴⁸ + S₃³³⁷ + S₃³⁰S₅²⁴ + S₃⁵⁰S₅¹² + S₃⁶³S₅²² + S₃⁶⁸S₅¹⁹ + S₃⁷³S₅¹⁶ + S₃⁷⁸S₅¹³ + S₃⁹⁶S₅²⁰ + S₃¹¹⁶S₅⁸ + S₃¹²⁴S₅²¹ + S₃¹⁴⁴S₅⁹ + S₃¹⁶²S₅¹⁶ + S₃¹⁸²S₅⁴ + S₃¹⁹⁰S₅¹⁷ + S₃²¹⁰S₅⁵ + S₃²²⁸S₅¹² + S₃²⁵⁶S₅¹³ + S₃³²²S₅⁹ + S₃³²⁷S₅⁶ + S₃³³²S₅³ + S₃²³³S₉⁵ + S₃²³⁶S₉⁴ + S₃³²⁸S₉³ + S₃³³¹S₉² + S₁₁S₃²⁸S₅²³ + S₁₁S₃³³S₅²⁰ + S₁₁S₃³⁸S₅¹⁷ + S₁₁S₃⁴³S₅¹⁴ + S₁₁S₃⁶¹S₅²¹ + S₁₁S₃⁶⁶S₅¹⁸ + S₁₁S₃⁷¹S₅¹⁵ + S₁₁S₃⁷⁶S₅¹² + S₁₁S₃⁹⁴S₅¹⁹ + S₁₁S₃⁹⁹S₅¹⁶ + S₁₁S₃¹⁰⁴S₅¹³ + S₁₁S₃¹⁰⁹S₅¹⁰ + S₁₁S₃¹²⁷S₅¹⁷ + S₁₁S₃¹³²S₅¹⁴ + S₁₁S₃¹³⁷S₅¹¹ + S₁₁S₃¹⁴²S₅⁸ + S₁₁S₃¹⁶⁰S₅¹⁵ + S₁₁S₃¹⁶⁵S₅¹² + S₁₁S₃¹⁷⁰S₅⁹ + S₁₁S₃¹⁷⁵S₅⁶ + S₁₁S₃¹⁹³S₅¹³ + S₁₁S₃¹⁹⁸S₅¹⁰ + S₁₁S₃²⁰³S₅⁷ + S₁₁S₃²⁰⁸S₅⁴ + S₁₁S₃²²⁶S₅¹¹ + S₁₁S₃²³¹S₅⁸ + S₁₁S₃²³⁶S₅⁵ + S₁₁S₃²⁴¹S₅² + S₁₁S₃²⁵⁹S₅⁹ + S₁₁S₃²⁶⁴S₅⁶ + S₁₁S₃²⁶⁹S₅³ + S₁₁S₃²⁶⁵S₉³ + S₁₁S₃²⁶⁸S₉² + S₃³²S₅²¹S₉ + S₃³⁷S₅¹⁸S₉ + S₃⁴²S₅¹⁵S₉ + S₃⁴⁷S₅¹²S₉ + S₃⁶⁵S₅¹⁹S₉ + S₃⁷⁵S₅¹³S₉ + S₃⁹⁸S₅¹⁷S₉ + S₃¹⁰³S₅¹⁴S₉ + S₃¹⁰⁸S₅¹¹S₉ + S₃¹¹³S₅⁸S₉ + S₃¹²⁶S₅¹⁸S₉ + S₃¹³¹S₅¹⁵S₉ + S₃¹³⁶S₅¹²S₉ + S₃¹⁴¹S₅⁹S₉ + S₃¹⁶⁴S₅¹³S₉ + S₃¹⁶⁹S₅¹⁰S₉ + S₃¹⁷⁴S₅⁷S₉ + S₃¹⁷⁹S₅⁴S₉ + S₃¹⁹²S₅¹⁴S₉ + S₃¹⁹⁷S₅¹¹S₉ + S₃²⁰²S₅⁸S₉ + S₃²⁰⁷S₅⁵S₉ + S₃²³⁰S₅⁹S₉ + S₃²³⁵S₅⁶S₉ + S₃²⁴⁰S₅³S₉ + S₃²⁵⁸S₅¹⁰S₉ + S₃²⁶³S₅⁷S₉ + S₃²⁶⁷S₅S₉³ + S₃²⁶⁸S₅⁴S₉ + S₃²⁷⁰S₅S₉² + S₃³²⁴S₅⁶S₉ + S₃²⁹S₅²¹S₉² + S₃³¹S₅¹⁸S₉³ + S₃³³S₅¹⁵S₉⁴ + S₃³⁵S₅¹²S₉⁵ + S₃³⁶S₅¹⁵S₉³ + S₃³⁸S₅¹²S₉⁴ + S₃³⁹S₅¹⁵S₉² + S₃⁶⁷S₅¹⁶S₉² + S₃⁶⁹S₅¹³S₉³ + S₃⁷²S₅¹³S₉² + S₃⁹⁵S₅¹⁷S₉² + S₃⁹⁷S₅¹⁴S₉³ + S₃⁹⁹S₅¹¹S₉⁴ + S₃¹⁰¹S₅⁸S₉⁵ + S₃¹⁰²S₅¹¹S₉³ + S₃¹⁰⁴S₅⁸S₉⁴ + S₃¹⁰⁵S₅¹¹S₉² + S₃¹²⁸S₅¹⁵S₉² + S₃¹³⁰S₅¹²S₉³ + S₃¹³⁵S₅⁹S₉³ + S₃¹³⁸S₅⁹S₉² + S₃¹⁶¹S₅¹³S₉² + S₃¹⁶³S₅¹⁰S₉³ + S₃¹⁶⁵S₅⁷S₉⁴ + S₃¹⁶⁷S₅⁴S₉⁵ + S₃¹⁶⁸S₅⁷S₉³ + S₃¹⁷⁰S₅⁴S₉⁴ + S₃¹⁷¹S₅⁷S₉² + S₃¹⁹⁴S₅¹¹S₉² + S₃¹⁹⁶S₅⁸S₉³ + S₃²⁰¹S₅⁵S₉³ + S₃²⁰⁴S₅⁵S₉² + S₃²²⁷S₅⁹S₉² + S₃²²⁹S₅⁶S₉³ + S₃²³¹S₅³S₉⁴ + S₃²³⁴S₅³S₉³ + S₃²³⁷S₅³S₉² + S₃²⁶⁰S₅⁷S₉² + S₃²⁶²S₅⁴S₉³ + S₃³²⁶S₅³S₉² + S₁₁S₃²⁷¹S₉ + S₃²⁷³S₅S₉ + S₁₁S₃³⁰S₅²⁰S₉ + S₁₁S₃⁴⁰S₅¹⁴S₉ + S₁₁S₃⁶³S₅¹⁸S₉ + S₁₁S₃⁷³S₅¹²S₉ + S₁₁S₃⁹⁶S₅¹⁶S₉ + S₁₁S₃¹⁰⁶S₅¹⁰S₉ + S₁₁S₃¹²⁹S₅¹⁴S₉ + S₁₁S₃¹³⁹S₅⁸S₉ + S₁₁S₃¹⁶²S₅¹²S₉ + S₁₁S₃¹⁷²S₅⁶S₉ + S₁₁S₃¹⁹⁵S₅¹⁰S₉ + S₁₁S₃²⁰⁵S₅⁴S₉ + S₁₁S₃²²⁸S₅⁸S₉ + S₁₁S₃²³⁸S₅²S₉ + S₁₁S₃²⁶¹S₅⁶S₉ + S₁₁S₃³²S₅¹⁷S₉² + S₁₁S₃³⁴S₅¹⁴S₉³ + S₁₁S₃³⁷S₅¹⁴S₉² + S₁₁S₃⁶⁵S₅¹⁵S₉² + S₁₁S₃⁶⁷S₅¹²S₉³ + S₁₁S₃⁷⁰S₅¹²S₉² + S₁₁S₃⁹⁸S₅¹³S₉² + S₁₁S₃¹⁰⁰S₅¹⁰S₉³ + S₁₁S₃¹⁰³S₅¹⁰S₉² + S₁₁S₃¹³¹S₅¹¹S₉² + S₁₁S₃¹³³S₅⁸S₉³ + S₁₁S₃¹³⁶S₅⁸S₉² + S₁₁S₃¹⁶⁴S₅⁹S₉² + S₁₁S₃¹⁶⁶S₅⁶S₉³ + S₁₁S₃¹⁶⁹S₅⁶S₉² + S₁₁S₃¹⁹⁷S₅⁷S₉² + S₁₁S₃¹⁹⁹S₅⁴S₉³ + S₁₁S₃²⁰²S₅⁴S₉² + S₁₁S₃²³⁰S₅⁵S₉² + S₁₁S₃²³²S₅²S₉³ + S₁₁S₃²³⁵S₅²S₉² + S₁₁S₃²⁶³S₅³S₉² + (S₁₁²S₃²⁷¹ + S₁₁²S₃²⁶⁵S₉² + S₁₁²S₃²⁶¹S₅⁶ + S₁₁²S₃²³⁸S₅² + S₁₁²S₃²³²S₅²S₉² + S₁₁²S₃²²⁸S₅⁸ + S₁₁²S₃²⁰⁵S₅⁴ + S₁₁²S₃¹⁹⁹S₅⁴S₉² + S₁₁²S₃¹⁹⁵S₅¹⁰ + S₁₁²S₃¹⁷²S₅⁶ + S₁₁²S₃¹⁶⁶S₅⁶S₉² + S₁₁²S₃¹⁶²S₅¹² + S₁₁²S₃¹³⁹S₅⁸ + S₁₁²S₃¹³³S₅⁸S₉² + S₁₁²S₃¹²⁹S₅¹⁴ + S₁₁²S₃¹⁰⁶S₅¹⁰ + S₁₁²S₃¹⁰⁰S₅¹⁰S₉² + S₁₁²S₃⁹⁶S₅¹⁶ + S₁₁²S₃⁷³S₅¹² + S₁₁²S₃⁶⁷S₅¹²S₉² + S₁₁²S₃⁶³S₅¹⁸ + S₁₁²S₃⁴⁰S₅¹⁴ + S₁₁²S₃³⁴S₅¹⁴S₉² + S₁₁²S₃³⁰S₅²⁰ + S₁₁S₃²⁷³S₅ + S₁₁S₃²⁷⁰S₅S₉ + S₁₁S₃²⁶⁸S₅⁴ + S₁₁S₃²⁶⁷S₅S₉² + S₁₁S₃²⁶⁴S₅S₉³ + S₁₁S₃²⁶³S₅⁷ + S₁₁S₃²⁶²S₅⁴S₉² + 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S₁₁S₃³⁵S₅¹²S₉⁴ + S₁₁S₃³³S₅¹⁵S₉³ + S₁₁S₃³²S₅²¹ + S₁₁S₃³¹S₅¹⁸S₉² + S₁₁S₃²⁹S₅²¹S₉ + S₃³⁹⁷ + S₃³⁹¹S₉² + S₃³⁸⁷S₅⁶ + S₃³⁶⁴S₅² + S₃³⁵⁸S₅²S₉² + S₃³⁵⁴S₅⁸ + S₃³³⁶S₅ + S₃³³³S₅S₉ + S₃³³¹S₅⁴ + S₃³³⁰S₅S₉² + S₃³²⁷S₅S₉³ + S₃³²⁶S₅⁷ + S₃³²⁵S₅⁴S₉² + S₃³²³S₅⁷S₉ + S₃³²¹S₅¹⁰ + S₃³⁰⁸ + S₃²⁹⁶S₉⁴ + S₃²⁸⁸S₅¹² + S₃²⁷⁵S₅² + S₃²⁷²S₅²S₉ + S₃²⁶⁷S₅⁵S₉ + S₃²⁶⁶S₅²S₉³ + S₃²⁶³S₅²S₉⁴ + S₃²⁶²S₅⁸S₉ + S₃²⁶¹S₅⁵S₉³ + S₃²⁵⁷S₅¹¹S₉ + S₃²⁵⁵S₅¹⁴ + S₃²⁴⁴S₅S₉ + S₃²⁴²S₅⁴ + S₃²³⁹S₅⁴S₉ + S₃²³⁴S₅⁷S₉ + S₃²³³S₅⁴S₉³ + S₃²³²S₅S₉⁵ + S₃²³⁰S₅⁴S₉⁴ + S₃²²⁹S₅¹⁰S₉ + S₃²²⁸S₅⁷S₉³ + S₃²²²S₅¹⁶ + S₃²⁰⁹S₅⁶ + S₃²⁰⁶S₅⁶S₉ + S₃²⁰¹S₅⁹S₉ + S₃²⁰⁰S₅⁶S₉³ + S₃¹⁹⁷S₅⁶S₉⁴ + S₃¹⁹⁶S₅¹²S₉ + S₃¹⁹⁵S₅⁹S₉³ + S₃¹⁹¹S₅¹⁵S₉ + S₃¹⁸⁹S₅¹⁸ + S₃¹⁷⁸S₅⁵S₉ + S₃¹⁷⁶S₅⁸ + S₃¹⁷³S₅⁸S₉ + S₃¹⁶⁸S₅¹¹S₉ + S₃¹⁶⁷S₅⁸S₉³ + S₃¹⁶⁶S₅⁵S₉⁵ + S₃¹⁶⁴S₅⁸S₉⁴ + S₃¹⁶³S₅¹⁴S₉ + S₃¹⁶²S₅¹¹S₉³ + S₃¹⁵⁶S₅²⁰ + S₃¹⁴³S₅¹⁰ + S₃¹⁴⁰S₅¹⁰S₉ + S₃¹³⁵S₅¹³S₉ + S₃¹³⁴S₅¹⁰S₉³ + S₃¹³³S₅¹⁶ + S₃¹³¹S₅¹⁰S₉⁴ + S₃¹³⁰S₅¹⁶S₉ + S₃¹²⁹S₅¹³S₉³ + S₃¹²⁷S₅¹⁶S₉² + S₃¹²⁵S₅¹⁹S₉ + S₃¹¹²S₅⁹S₉ + S₃¹¹⁰S₅¹² + S₃¹⁰⁷S₅¹²S₉ + S₃¹⁰²S₅¹⁵S₉ + S₃¹⁰¹S₅¹²S₉³ + 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+ S₁₁²S₃⁹⁵S₅¹⁷ + S₁₁²S₃⁷²S₅¹³ + S₁₁²S₃⁶⁶S₅¹³S₉² + S₁₁²S₃⁶²S₅¹⁹ + S₁₁²S₃³⁹S₅¹⁵ + S₁₁²S₃³³S₅¹⁵S₉² + S₁₁²S₃²⁹S₅²¹ + S₁₁S₃³⁰⁵ + S₁₁S₃³⁰²S₉ + S₁₁S₃³⁰⁰S₅³ + S₁₁S₃²⁹⁹S₉² + S₁₁S₃²⁹⁶S₉³ + S₁₁S₃²⁹⁵S₅⁶ + S₁₁S₃²⁹⁴S₅³S₉² + S₁₁S₃²⁹²S₅⁶S₉ + S₁₁S₃²⁹⁰S₅⁹ + S₁₁S₃²⁴⁴S₅ + S₁₁S₃²³²S₅S₉⁴ + S₁₁S₃²²⁴S₅¹³ + S₁₁S₃¹⁷⁸S₅⁵ + S₁₁S₃¹⁶⁶S₅⁵S₉⁴ + S₁₁S₃¹⁵⁸S₅¹⁷ + S₁₁S₃¹¹²S₅⁹ + S₁₁S₃¹⁰⁰S₅⁹S₉⁴ + S₁₁S₃⁹²S₅²¹ + S₁₁S₃⁴⁶S₅¹³ + S₁₁S₃⁴¹S₅¹⁶ + S₁₁S₃³⁸S₅¹⁶S₉ + S₁₁S₃³⁶S₅¹⁹ + S₁₁S₃³⁵S₅¹⁶S₉² + S₁₁S₃³⁴S₅¹³S₉⁴ + S₁₁S₃³²S₅¹⁶S₉³ + S₁₁S₃³¹S₅²² + S₁₁S₃³⁰S₅¹⁹S₉² + S₁₁S₃²⁸S₅²²S₉ + S₃³⁹⁶S₅ + S₃³⁹⁰S₅S₉² + S₃³⁸⁶S₅⁷ + S₃³⁶⁸ + S₃³⁶⁵S₉ + S₃³⁶²S₉² + S₃³⁵⁹S₉³ + S₃³⁵⁸S₅⁶ + S₃³⁵⁵S₅⁶S₉ + S₃³³⁵S₅² + S₃³³²S₅²S₉ + S₃³³⁰S₅⁵ + S₃³²⁹S₅²S₉² + S₃³²⁶S₅²S₉³ + S₃³²⁵S₅⁸ + S₃³²⁴S₅⁵S₉² + S₃³²²S₅⁸S₉ + S₃³²⁰S₅¹¹ + S₃³⁰⁴S₅S₉ + S₃³⁰¹S₅S₉² + S₃²⁹⁹S₅⁴S₉ + S₃²⁹⁸S₅S₉³ + S₃²⁹⁵S₅S₉⁴ + S₃²⁹⁴S₅⁷S₉ + S₃²⁹³S₅⁴S₉³ + S₃²⁹¹S₅⁷S₉² + S₃²⁸⁹S₅¹⁰S₉ + S₃²⁷⁹ + S₃²⁷⁶S₉ + S₃²⁷⁴S₅³ + S₃²⁶⁸S₅³S₉² + S₃²⁶⁷S₉⁴ + S₃²⁶⁴S₉⁵ + S₃²⁵⁹S₅¹² + S₃²⁵⁸S₅⁹S₉² + S₃²⁵⁶S₅¹²S₉ + S₃²⁵⁴S₅¹⁵ + S₃²⁴³S₅²S₉ + 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S₁₁²S₃³⁸S₅¹⁶ + S₁₁²S₃³²S₅¹⁶S₉² + S₁₁²S₃²⁸S₅²² + S₃⁴²⁸ + S₃⁴²²S₉² + S₃⁴¹⁸S₅⁶ + S₃³⁶²S₅⁴ + S₃³⁵⁶S₅⁴S₉² + S₃³⁵²S₅¹⁰ + S₃³⁰⁶S₅² + S₃³⁰⁰S₅²S₉² + S₃²⁹⁰S₅⁸S₉² + S₃²⁸⁶S₅¹⁴ + S₃²⁵⁰ + S₃²⁴⁴S₉² + S₃²⁴⁰S₅⁶ + S₃²³⁸S₉⁴ + S₃²³²S₉⁶ + S₃²³⁰S₅¹² + S₃²²⁸S₅⁶S₉⁴ + S₃²²⁴S₅¹²S₉² + S₃²²⁰S₅¹⁸ + S₃¹⁸⁴S₅⁴ + S₃¹⁷⁸S₅⁴S₉² + S₃¹⁷⁴S₅¹⁰ + S₃¹⁷²S₅⁴S₉⁴ + S₃¹⁶⁶S₅⁴S₉⁶ + S₃¹⁶²S₅¹⁰S₉⁴ + S₃¹¹⁸S₅⁸ + S₃¹¹²S₅⁸S₉² + S₃¹⁰⁸S₅¹⁴ + S₃¹⁰⁶S₅⁸S₉⁴ + S₃¹⁰⁰S₅⁸S₉⁶ + S₃⁹⁶S₅¹⁴S₉⁴ + S₃⁵²S₅¹² + S₃⁴⁶S₅¹²S₉² + S₃⁴⁰S₅¹²S₉⁴ + S₃³⁶S₅¹⁸S₉² + S₃³⁴S₅¹²S₉⁶ + S₃³²S₅²⁴ + S₃³⁰S₅¹⁸S₉⁴)x⁶.

4. By (10), δ⁽¹²⁾ = 0. Then, compute λ⁽¹²⁾(x) = xλ⁽¹¹⁾(x), l⁽¹²⁾ = l⁽¹¹⁾ = 6, and γ⁽¹²⁾ = γ⁽¹¹⁾, respectively.

2. Set k = 13. Because k = 13 > 2t = 12, the recursion stops.

Finally, the error-locator polynomial μ⁽¹²⁾(x) = L₆(x) of degree 6 is obtained. By applying Chien search method, the roots of L₆(x) are exactly the inverse of the six error locators {0, 1, 2, 3, 15, 26}. A C++ program shows that the total weight-6 error patterns with S₁ = 0 can be corrected.