# New Soft Set Based Class of Linear Algebraic Codes

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

_{m}[8,9]. However fuzzy codes and distance properties was developed by [10]. For literature used in this paper on coding theory, see Reference [11].

## 2. Fundamental Notions

#### 2.1. Algebraic Linear Codes and Their Properties

_{2}= {0, 1}, the finite field of characteristic two. We use F to denote Z

_{2}.

**Definition**

**1.**

- 1.
- V is a commutative group under addition.
- 2.
- For any element a in F and any element v in V, a.v = v.a is in V.
- 3.
- Distributive law: For any u and v in V and for any a,b ∈ Fa.(u + v) = a.u + a.v; (a + b).v = a.v + b.v.
- 4.
- Associative law: For any v in V and any a and b in F; (a.b).v = a.(b.v).
- 5.
- Let 1 be the unit element of F. Then for any v in V, 1.v = v and 0.v = 0 for 0 ∈ F and ‘$\overline{0}$’ is the zero vector of V.We call a proper subset U of V (U ⊂ V) to be a vector subspace of V over F if U itself is a vector space over F.

**Definition**

**2.**

^{k}codewords is called a linear code, denoted by C (n, k), if and only if its 2

^{k}codewords form a k-dimensional subspace of the vector space V

^{n}of all the n tuples over the field GF(2).

_{i,j}∈ Z

_{2}= F; for 0 ≤ i ≤ k − 1 and 0 ≤ j ≤ n − 1. Consider u = (u

_{0}u

_{1}… u

_{k−1}), the message to be encoded, the corresponding codeword v is given by v = u.G. Every codeword v in C (n, k) is a linear combination of k codewords.

_{k × k}), then parity check matrix H can be got in the standard form as H = (I

_{n−k × n−k}; A

^{T}). The generator matrix can be in any other form, and then the parity check matrix can be found out by the usual methods given in Reference [11].

^{T}is obtained from the parity check matrix H. Thus, the parity check matrix H of a code helps to detect the error from the received word. The error correcting capacity of a code depends on the metric that is used over the code. The most basic metric, namely the Hamming metric of the code is defined as follows:

**Definition**

**3.**

_{1}… x

_{n}) and y = (y

_{1}… y

_{n}) in V

^{n}, the n dimensional vector space over the field F = Z

_{2}, the Hamming distance d(x, y) and the Hamming weight w(x) are defined as follows:

_{i}:x

_{i}≠ y

_{i}; x

_{i}∈ x; y

_{i}∈ y}|

_{i}:x

_{i}≠ 0; x

_{i}∈ x}|.

**Definition**

**4.**

_{min}of a code C(n, k) is defined as

#### 2.2. Soft Set Theory

**Definition**

**5.**

## 3. Algebraic Soft Linear Codes and Their Properties

**Definition**

**6.**

_{2}; be the field of characteristic two. Let W = F × … × F = F

^{m}, be a vector space over the field F of dimension m. P(W) be the power set of W. (f, D) is said to be a soft algebraic linear code over F if and only if f(d) is a linear algebraic code of W for all d ∈ D; D ⊂ V, where V is the set of parameters.

**Example**

**1.**

^{3}be a vector space over the field F. (f, D) is a soft linear code over W where f(D) = {f(d

_{1}), f(d

_{2})} with

_{1}) = {000, 111} and f(d

_{2}) = {000, 110, 101, 011}.

**Definition**

**7.**

^{m}, be a vector space over the field F of dimension m. P(W) be the power set of W. (f, D) be a soft algebraic linear code over F. Let f(D) = {f(d

_{1}), …, f(d

_{t})} where each f(d

_{i}); 1 ≤ i ≤ t is a linear algebraic code of W. Each t-tuple {x

_{1}, x

_{2}, …, x

_{t}}; x

_{i}∈ f(d

_{i}); 1 ≤ i ≤ t is defined as the soft codeword of the soft algebraic code (f, D). We have |f(d

_{1})| × |f(d

_{2})|× … × |f(d

_{t})| number of soft codewords for this (f, D).

_{1}) and f(d

_{2}), respectively.

**Definition**

**8.**

^{m}, be a vector space over the field F of dimension m. (f, D) be a soft algebraic linear code over F. Let f(D) = {f(d

_{1}), …, f(d

_{t})} where each f(d

_{i}); 1 ≤ i ≤ t is a linear algebraic code of W. Here each f(d

_{i}) ∈ f(D) is a linear algebraic code and dimension of f(d

_{i}) is n

_{i}where n

_{i}is the number of linear independent elements of f(d

_{i}). The soft dimension of (f, D) = {n

_{1}, …, n

_{t}} and the number of soft codewords of (f, D) is |f(d

_{1})| × |f(d

_{2})| × … × |f(d

_{t}) |where 1 ≤ i ≤ t.

**Definition**

**9.**

_{1}, n

_{2}, …, n

_{t}} is such that n

_{1}= n

_{2}= … = n

_{t}.

**Example**

**2.**

^{5}over the field F. Consider

_{1}) = {00000, 11111, 10110, 01001},

f(d

_{2}) = {00000, 11111, 11001, 00110},

f(d

_{3}) = {00000, 11111, 00111, 11000}, and

f(d

_{4}) = {00000, 11111, 11100, 00011}.

**Theorem**

**1.**

**Proof.**

_{1}) and f(d

_{2}) are different. □

**Definition**

**10.**

_{1}), …, f(d

_{t})}. We know that associated with each f(d

_{i}) we have an algebraic code of dimension n

_{i}. Let G

_{i}; 1 ≤ i ≤ t be the generator matrix associated with this algebraic code associated with f(d

_{i}). Then we define the soft generator matrix G

_{s}as the t-matrix given by G

_{s}= [G

_{1}|G

_{2}| … |G

_{t}]. If the each generator matrix G

_{i}of the soft generator matrix G

_{s}is represented in the standard form then the soft generator matrix G

_{s}is known as soft canonical generator matrix and is denoted by G

_{s}*.

**Example**

**3.**

_{i}is the generator matrix of the algebraic code associated with f(d

_{i}); i = 1, 2, 3, 4; clearly this G

_{S}is not the soft canonical generator matrix.

**Example**

**4.**

^{5}, where

_{1}) = {00000, 10010, 01001, 00110, 11011, 10100, 01111, 11101};

f(d

_{2}) = {00000, 11111, 10110, 01001}

_{1}and G

_{2}where

**Definition**

**11.**

_{1}), …, f(d

_{t})} where each f(d

_{i}) is linear algebraic code, let H

_{i}(1 ≤ i ≤ t) be the parity check matrix associated with each linear algebraic code. Then H

_{S}= {H

_{1}|H

_{2}| … |H

_{t}} is the soft parity check matrix associated with the soft linear algebraic code.

_{i}is taken in the standard form then the corresponding soft parity check matrix${H}_{s}^{*}$is defined as the soft canonical parity check matrix of the soft algebraic linear code.

## 4. Soft Linear Algebraic Decoding Algorithms

_{i}’s are coset leaders. Syndrome of e

_{i}, s(e

_{i}) = e

_{i}H

^{T}; 0 ≤ i ≤ t.

**Definition**

**12.**

_{s}= (H

_{1}|H

_{2}| … |H

_{t}) be the soft parity check matrix of (f, D). Suppose y is the received soft message, the soft syndrome of y is defined as s(y) = y${H}_{s}^{T}$; if s(y) ≠ (o) then we say the soft codeword has soft error.

^{m}be a vector space of dimension n over F = Z

_{2}. Let (f, D) be a soft algebraic code with f(D) = (f(d

_{1}), …, f(d

_{t})) where each f(d

_{i}); 1 ≤ i ≤ t; is a linear algebraic code over W. Any soft codeword in (f, D) will be of the form x = (x

_{1}, x

_{2}, …, x

_{t}) where x

_{i}∈ f(d

_{i}) and x

_{i}= $\left({y}_{1}^{i},\dots ,{y}_{m}^{i}\right)$ a m-tuple for which it will have k

_{i}message symbols; 1 ≤ i ≤ t.

^{T}= $z\left({H}_{1}^{T}|{H}_{2}^{T}|\dots |{H}_{t}^{T}\right)$ where each H

_{i}is the parity check matrix of the linear algebraic code associated with f(d

_{i}); 1 ≤ i ≤ t.

**Example**

**5.**

_{0}= {000, 000}, e

_{1}= {100, 100}, e

_{2}= {010, 010} and e

_{3}= {001, 001}

^{T}= {01, 1} using the table, it corresponds to soft coset leader e

_{3}. This y = {110, 010} appears as the last one in the table in the middle column. Thus, the error term is e

_{3}= {001, 001}, which should be added with y. Now, y + e

_{3}= {110, 010} + {001, 001} = {111, 011} ∈ f(D), which is the correct codeword.

**Theorem**

**2.**

_{1}, …, y

_{t}); y

_{i}∈ W = F

^{m}; 1 ≤ i ≤ t; then there is a soft codeword nearest to y given by x = y + soft coset leader e

_{i}of the soft code (f, D).

**Proof.**

_{1}, …, y

_{t}) be the received codeword, we find the soft syndrome;

_{i}from the collection of coset leaders using the one analogues Table 1. Then, for all soft coset leaders we calculate the soft set-based syndrome and make a table of soft linear algebraic coset leaders with their soft set-based syndromes. For decoding a soft linear algebraic codeword y, we can merely find the soft set-based syndrome of the soft linear algebraic codeword and then compare soft coset leader syndrome with their soft set-based syndrome. After the comparison, we add the soft decoded word to the soft linear algebraic coset leader. Thus, y is soft decoded as x = y + e

_{i}; e

_{i}is the soft coset leader and x is the corrected word. □

## 5. Soft Set-Based Communication Transmission and Comparison of Soft Linear Algebraic Codes and Linear Algebraic Codes

_{1}, …, a

_{m}), there are m parameters. A soft set-based communication transmission reduces to classical communication transmission if we have m = 1. The model of soft set-based communication transmission is given in the following Figure 1.

_{i}” can have some attribute that can trick the hackers. Therefore, the soft linear codes can be more secure as compared to the classical linear codes due to the parameterization. The soft linear codes have a different distinct structure. Soft linear code is a collection of subspaces, whereas a linear code is only one subspace. Each subspace relies on the set of parameters that are used. Hence, soft linear codes are more generalized in comparison to the linear codes.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Shannon, C.E. A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Commun. Rev.
**2001**, 5, 3–55. [Google Scholar] [CrossRef] - Shannon, C.E. Certain results in coding theory for noisy channels. Inf. Control
**1957**, 1, 6–25. [Google Scholar] [CrossRef] - Hocquenghem, A. Codes correcteurs d’erreurs. Chiffres
**1959**, 2, 147–156. (In French) [Google Scholar] - Bose, R.C.; Ray-Chaudhuri, D.K. On A Class of Error Correcting Binary Group Codes. Inf. Control
**1960**, 3, 68–79. [Google Scholar] [CrossRef] - Conway, J.H.; Sloane, N.J.A. Self-dual codes over the integers modulo 4. J. Comb. Theory Ser. A
**1993**, 62, 30–45. [Google Scholar] [CrossRef] - Dougherty, S.T.; Shiromoto, K. Maximum distance codes over rings of order 4. IEEE Trans. Inf. Theory
**2001**, 47, 400–404. [Google Scholar] [CrossRef] - Norton, G.H.; Salagean, A. On the Hamming distance of linear codes over a finite chain ring. IEEE Trans. Inf. Theory
**2000**, 46, 1060–1067. [Google Scholar] [CrossRef] [Green Version] - Spiegel, E. Codes over Zm. Inf. Control
**1977**, 35, 48–51. [Google Scholar] [CrossRef] - Spiegel, E. Codes over Zm, Revisited. Inf. Control
**1978**, 37, 100–104. [Google Scholar] [CrossRef] - Von Kaenel, P.A. Fuzzy codes and distance properties. Fuzzy Sets Syst.
**1982**, 8, 199–204. [Google Scholar] [CrossRef] - Lidl, R.; Pilz, G. Applied Abstract Algebra; Springer: New York, NY, USA, 1984. [Google Scholar]
- Molodtsov, D. Soft set theory—First results. Comput. Math. Appl.
**1999**, 37, 19–31. [Google Scholar] [CrossRef] [Green Version] - Zadeh, L.A. Information and control. Fuzzy Sets
**1965**, 8, 338–353. [Google Scholar] - Aktaş, H.; Çağman, N. Soft sets and soft groups. Inf. Sci.
**2007**, 177, 2726–2735. [Google Scholar] [CrossRef] - Maji, P.K. Neutrosophic Soft Set. Ann. Fuzzy Math. Inform.
**2013**, 5, 157–168. [Google Scholar] - Shabir, M.; Ali, M.; Naz, M.; Smarandache, F. Soft neutrosophic group. Neutrosophic Sets Syst.
**2013**, 1, 13–25. [Google Scholar] - Smarandache, F.; Ali, M.; Shabir, M. Soft Neutrosophic Algebraic Structures and Their Generalization. arXiv, 2014; arXiv:1408.5507. [Google Scholar]
- Ali, M.; Dyer, C.; Shabir, M.; Smarandache, F. Soft neutrosophic loops and their generalization. Neutrosophic Sets Syst.
**2014**, 4, 55–75. [Google Scholar] - Ali, M.; Smarandache, F.; Shabir, M.; Naz, M. Soft neutrosophic ring and soft neutrosophic field. Neutrosophic Sets Syst.
**2014**, 3, 55–61. [Google Scholar] - Ali, M.I.; Feng, F.; Liu, X.; Min, W.K.; Shabir, M. On some new operations in soft set theory. Comput. Math. Appl.
**2009**, 57, 1547–1553. [Google Scholar] [CrossRef] [Green Version] - Alcantud, J.C.R. Some formal relationships among soft sets, fuzzy sets, and their extensions. Int. J. Approx. Reason.
**2016**, 68, 45–53. [Google Scholar] [CrossRef] - Vasantha, W.B.; Selvaraj, R.S. Multi-covering radii of codes with rank metric. In Proceedings of the 2002 IEEE Information Theory Workshop (ITW 2002), Bangalore, India, 25 October 2002. [Google Scholar] [CrossRef]
- Vasantha, W.B.; Raja Durai, R.S. T-direct codes: An application to T-user BAC. In Proceedings of the 2002 IEEE Information Theory Workshop (ITW 2002), Bangalore, India, 25 October 2002. [Google Scholar] [CrossRef]
- Fatimah, F.; Rosadi, D.; Hakim, R.F.; Alcantud, J.C.R. N-soft sets and their decision making algorithms. Soft Comput.
**2018**, 22, 3829–3842. [Google Scholar] [CrossRef] - Tuan, T.M.; Chuan, P.M.; Ali, M.; Ngan, T.T.; Mittal, M.; Son, L.H. Fuzzy and neutrosophic modeling for link prediction in social networks. Evol. Syst.
**2018**, 1–6. [Google Scholar] [CrossRef] - Dey, A.; Son, L.; Kumar, P.; Selvachandran, G.; Quek, S. New Concepts on Vertex and Edge Coloring of Simple Vague Graphs. Symmetry
**2018**, 10, 373. [Google Scholar] [CrossRef] - Khan, M.; Son, L.; Ali, M.; Chau, H.; Na, N.; Smarandache, F. Systematic review of decision making algorithms in extended neutrosophic sets. Symmetry
**2018**, 10, 314. [Google Scholar] [CrossRef] - Son, L.H.; Fujita, H. Neural-fuzzy with representative sets for prediction of student performance. Appl. Intell.
**2018**, 1–16. [Google Scholar] [CrossRef] - Jha, S.; Kumar, R.; Chatterjee, J.M.; Khari, M.; Yadav, N.; Smarandache, F. Neutrosophic soft set decision making for stock trending analysis. Evol. Syst.
**2018**, 1–7. [Google Scholar] [CrossRef] - Ngan, R.T.; Son, L.H.; Cuong, B.C.; Ali, M. H-max distance measure of intuitionistic fuzzy sets in decision making. Appl. Soft Comput.
**2018**, 69, 393–425. [Google Scholar] [CrossRef] - Ali, M.; Thanh, N.D.; Van Minh, N. A neutrosophic recommender system for medical diagnosis based on algebraic neutrosophic measures. Appl. Soft Comput.
**2018**, 71, 1054–1071. [Google Scholar] [CrossRef] [Green Version] - Ali, M.; Son, L.H.; Khan, M.; Tung, N.T. Segmentation of dental X-ray images in medical imaging using neutrosophic orthogonal matrices. Expert Syst. Appl.
**2018**, 91, 434–441. [Google Scholar] [CrossRef] - Ali, M.; Dat, L.Q.; Son, L.H.; Smarandache, F. Interval complex neutrosophic set: Formulation and applications in decision-making. Int. J. Fuzzy Syst.
**2018**, 20, 986–999. [Google Scholar] [CrossRef] - Nguyen, G.N.; Ashour, A.S.; Dey, N. A survey of the state-of-the-arts on neutrosophic sets in biomedical diagnoses. Int. J. Mach. Learn. Cybern.
**2017**, 1–13. [Google Scholar] [CrossRef] - Ngan, R.T.; Ali, M.; Son, L.H. δ-equality of intuitionistic fuzzy sets: A new proximity measure and applications in medical diagnosis. Appl. Intell.
**2018**, 48, 499–525. [Google Scholar] [CrossRef] - Ali, M.; Son, L.H.; Deli, I.; Tien, N.D. Bipolar neutrosophic soft sets and applications in decision making. J. Intell. Fuzzy Syst.
**2017**, 33, 4077–4087. [Google Scholar] [CrossRef]

Soft Coset Leaders | Soft Codewords as Cosets of (f, D) | Soft Syndromes |
---|---|---|

e_{0} = {000, 000} | {000, 000}, (000, 110}, {000, 101}, {000, 011}, {111, 000}, {111, 110}, {111, 101}, {111, 011} | e_{0}H^{T} = {00, 0} |

e_{1} = {100, 100} | {100, 100}, (100, 010}, {100, 001}, {100, 111}, {011, 100}, {111, 010}, {011, 001}, {011, 111} | e_{1}H^{T} = {10, 1} |

e_{2} = {010, 010} | {010, 010}, (010, 100}, {010, 111}, {010, 001}, {101, 010}, {101, 100}, {101, 111}, {101, 001} | e_{2}H^{T} = {11, 1} |

e_{3} = {001, 001} | {001, 001}, (001, 111}, {001, 100}, {001, 010}, {110, 001}, {110, 111}, {110, 100}, {110, 010} | e_{3}H^{T} = {01, 1} |

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**MDPI and ACS Style**

Ali, M.; Khan, H.; Son, L.H.; Smarandache, F.; Kandasamy, W.B.V.
New Soft Set Based Class of Linear Algebraic Codes. *Symmetry* **2018**, *10*, 510.
https://doi.org/10.3390/sym10100510

**AMA Style**

Ali M, Khan H, Son LH, Smarandache F, Kandasamy WBV.
New Soft Set Based Class of Linear Algebraic Codes. *Symmetry*. 2018; 10(10):510.
https://doi.org/10.3390/sym10100510

**Chicago/Turabian Style**

Ali, Mumtaz, Huma Khan, Le Hoang Son, Florentin Smarandache, and W. B. Vasantha Kandasamy.
2018. "New Soft Set Based Class of Linear Algebraic Codes" *Symmetry* 10, no. 10: 510.
https://doi.org/10.3390/sym10100510