Polynomial over a finite field

Polynomial $f(x)$ ${\ displaystyle f (x)}$ $f (x)$ over the final field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ $\ mathbb {F} _ {q}$ called the formal sum of the form

f(x)=f_{0}+f_{1}x+\ldots +f_{m}x^{m},\quad f_{i}\in \mathbb {F} _{q},\quad f_{m}\neq 0.

{\ displaystyle f (x) = f_ {0} + f_ {1} x + \ ldots + f_ {m} x ^ {m}, \ quad f_ {i} \ in \ mathbb {F} _ {q}, \ quad f_ {m} \ neq 0.}

{\ displaystyle f (x) = f_ {0} + f_ {1} x + \ ldots + f_ {m} x ^ {m}, \ quad f_ {i} \ in \ mathbb {F} _ {q}, \ quad f_ {m} \ neq 0.}

Here $m$ ${\ displaystyle m}$ $m$ Is a non-negative integer called the degree of the polynomial $f(x)$ ${\ displaystyle f (x)}$ $f (x)$ , but $x^{k},k\in \mathbb {N} _{0}$ ${\ displaystyle x ^ {k}, k \ in \ mathbb {N} _ {0}}$ $x ^ k, k \ in \ mathbb N_0$ Are elements of algebra over $\mathbb {F} _{q},$ ${\ displaystyle \ mathbb {F} _ {q},}$ $\ mathbb {F} _ {q},$ the multiplication of which is specified by the rules:

x^{k}\cdot x^{m}=x^{k+m},

{\ displaystyle x ^ {k} \ cdot x ^ {m} = x ^ {k + m},}

x ^ k \ cdot x ^ m = x ^ {k + m},

x^{0}\equiv 1.

{\ displaystyle x ^ {0} \ equiv 1.}

x ^ 0 \ equiv 1.

Such a definition allows us to formally multiply polynomials without worrying about the fact that different degrees of the same element of a finite field can coincide ^[1] ^[2] .

Any function over a finite field can be defined using some polynomial (for example, the Lagrange interpolation polynomial ).

Content

1 Related Definitions
2 Roots of a polynomial
3 Cyclotomy class
- 3.1 Examples of cyclotomic classes
- 3.2 Relations with the roots of polynomials
4 Types of polynomials
- 4.1 Primitive polynomials
- 4.2 Circular polynomials
- 4.3 Zhegalkin polynomials
5 Application
6 See also
7 notes
8 Literature

Related Definitions

Number $m\geqslant 0$ ${\ displaystyle m \ geqslant 0}$ $m\geqslant 0$ called the degree of the polynomial and is denoted as $\deg(f(x))$ ${\ displaystyle \ deg (f (x))}$ $\deg(f(x))$ ^[2] .
If $f_{m}=1$ ${\ displaystyle f_ {m} = 1}$ $f_m=1$ , then the polynomial is called normalized (reduced) ^[2] . A polynomial can always be normalized by dividing it by a coefficient $f_{m}$ ${\ displaystyle f_ {m}}$ $f_m$ with an older degree.
The sum and product of polynomials are defined in the usual way, and operations with coefficients are carried out in the field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ $\mathbb {F} _{q}$ .
For two polynomials $f$ $($ $x$ $)$ {\ displaystyle f (x)} and $h$ $($ $x$ $)$ $\neq$ $0$ {\ displaystyle h (x) \ neq 0} there are always polynomials $t$ $($ $x$ $)$ {\ displaystyle t (x)} and $r$ $($ $x$ $)$ {\ displaystyle r (x)} over the field $F$ $q$ {\ displaystyle \ mathbb {F} _ {q}} that the relation will be satisfied
$f(x)=t(x)h(x)+r(x).$ ${\ displaystyle f (x) = t (x) h (x) + r (x).}$ $f(x)=t(x)h(x)+r(x).$
- If the degree $r(x)$ ${\ displaystyle r (x)}$ $r(x)$ strictly less degree $h(x)$ ${\ displaystyle h (x)}$ $h(x)$ , then this relation is called the polynomial representation $f(x)$ ${\ displaystyle f (x)}$ $f(x)$ in the form of quotient and remainder of division $f(x)$ ${\ displaystyle f (x)}$ $f(x)$ on $h(x)$ ${\ displaystyle h (x)}$ $h(x)$ , and such a representation is unique ^[3] . It's clear that $f(x)-r(x)$ ${\ displaystyle f (x) -r (x)}$ $f(x)-r(x)$ divided without remainder into $h(x)$ ${\ displaystyle h (x)}$ $h(x)$ that is written as $f(x)\equiv r(x){\pmod {h(x)}}$ ${\ displaystyle f (x) \ equiv r (x) {\ pmod {h (x)}}}$ $f(x)\equiv r(x)\pmod{h(x)}$ ^[4] .
- If $r(x)=0$ ${\ displaystyle r (x) = 0}$ $r(x)=0$ then the polynomial $h(x)$ ${\ displaystyle h (x)}$ $h(x)$ called a polynomial divisor $f(x)$ ${\ displaystyle f (x)}$ $f(x)$ ^[5] .
A polynomial is irreducible over a field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ $\mathbb {F} _{q}$ if it has no non-trivial divisors (degrees greater than 0 and less $\deg(f(x))$ ${\ displaystyle \ deg (f (x))}$ $\deg(f(x))$ ) ^[5] ^[6] .
Field extension $GF(q)$ ${\ displaystyle GF (q)}$ $GF(q)$ called set $F[x]_{/(p(x))}$ ${\ displaystyle F [x] _ {/ (p (x))}}$ $F[x]_{/(p(x))}$ residue classes modulo an irreducible polynomial $p(x)$ ${\ displaystyle p (x)}$ $p(x)$ over the field $GF(q)$ ${\ displaystyle GF (q)}$ $GF(q)$ ^[6] .
Minimum polynomial (minimum function) for an element $\beta$ ${\ displaystyle \ beta}$ $\beta$ from the extended field such a normalized polynomial is called $m(x)$ ${\ displaystyle m (x)}$ $m(x)$ over $GF(q)$ ${\ displaystyle GF (q)}$ $GF(q)$ minimum degree that $m(\beta )=0$ ${\ displaystyle m (\ beta) = 0}$ $m(\beta)=0$ ^[7] ^[8] .
The root of a polynomial is any element of a field whose value of this polynomial is equal to zero.
Conjugate elements of a field that are the roots of the same irreducible polynomial are called conjugate ^[9] .

Polynomial Roots

A polynomial of degree m has exactly m roots (taking into account multiplicity) belonging to some extended field $\mathbb {F} _{Q},\quad \mathbb {F} _{q}\subseteq \mathbb {F} _{Q}$ ${\ displaystyle \ mathbb {F} _ {Q}, \ quad \ mathbb {F} _ {q} \ subseteq \ mathbb {F} _ {Q}}$ . If $q=p^{s}$ ${\ displaystyle q = p ^ {s}}$ where $p$ ${\ displaystyle p}$ - simple then $Q=p^{S},\;S\geqslant s$ ${\ displaystyle Q = p ^ {S}, \; S \ geqslant s}$ . Based on the properties of the finite fields, any element of the field $\mathbb {F} _{Q}$ ${\ displaystyle \ mathbb {F} _ {Q}}$ is the root of the binomial $x^{Q}-x$ ${\ displaystyle x ^ {Q} -x}$ :

x^{Q}-x=\prod _{\alpha \in \mathbb {F} _{Q}}{(x-\alpha )}

{\ displaystyle x ^ {Q} -x = \ prod _ {\ alpha \ in \ mathbb {F} _ {Q}} {(x- \ alpha)}}

So the roots of the polynomial $f(x)$ ${\ displaystyle f (x)}$ are also the roots of the binomial $x^{Q}-x$ ${\ displaystyle x ^ {Q} -x}$ ^[10] .

The Bezout theorem and its corollaries are valid:

Remainder of the division $f(x)$ ${\ displaystyle f (x)}$ on $(x-a)$ ${\ displaystyle (xa)}$ is equal to $f(a)$ ${\ displaystyle f (a)}$ .

If $x_{0}$ ${\ displaystyle x_ {0}}$ - root $f(x)$ ${\ displaystyle f (x)}$ then $(x-x_{0})$ ${\ displaystyle (x-x_ {0})}$ divides $f(x)$ ${\ displaystyle f (x)}$ .

If $x_{1},\;\ldots ,\;x_{m}$ ${\ displaystyle x_ {1}, \; \ ldots, \; x_ {m}}$ the essence of the roots $f(x)$ ${\ displaystyle f (x)}$ then

f(x)=a(x-x_{1})(x-x_{2})\ldots (x-x_{m}).

{\ displaystyle f (x) = a (x-x_ {1}) (x-x_ {2}) \ ldots (x-x_ {m}).}

The following theorems are also valid:

Theorem 1 If $x_{0}$ ${\ displaystyle x_ {0}}$ - root $f(x)$ ${\ displaystyle f (x)}$ then $x_{0}^{q}$ ${\ displaystyle x_ {0} ^ {q}}$ - also root $f(x)$ ${\ displaystyle f (x)}$ ^[11] .

Theorem 2 The conjugate elements of the Galois field are of the same order ^[9] .

Cyclotomy class

A consequence of Theorem 1 may be the fact that if $\alpha \in \mathbb {F} _{Q}$ ${\ displaystyle \ alpha \ in \ mathbb {F} _ {Q}}$ - polynomial root $f(x)$ ${\ displaystyle f (x)}$ over the field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ then $\alpha ^{q},\;\alpha ^{q^{2}},\;\alpha ^{q^{3}},\;\ldots \in \mathbb {F} _{Q}$ ${\ displaystyle \ alpha ^ {q}, \; \ alpha ^ {q ^ {2}}, \; \ alpha ^ {q ^ {3}}, \; \ ldots \ in \ mathbb {F} _ {Q }}$ are its roots.

Definition: cyclotomic class over a field $\mathbb {F} _{q},\;q=p^{s}$ ${\ displaystyle \ mathbb {F} _ {q}, \; q = p ^ {s}}$ generated by some element $\alpha \in \mathbb {F} _{Q},\;Q=p^{S}$ ${\ displaystyle \ alpha \ in \ mathbb {F} _ {Q}, \; Q = p ^ {S}}$ called the set of all the different elements $\mathbb {F} _{Q}$ ${\ displaystyle \ mathbb {F} _ {Q}}$ which are $q$ ${\ displaystyle q}$ degrees $\alpha$ ${\ displaystyle \ alpha}$ ^[12] .

If $\alpha$ ${\ displaystyle \ alpha}$ Is a primitive element ^[13] (such an element that $\alpha ^{Q-1}=1$ ${\ displaystyle \ alpha ^ {Q-1} = 1}$ and $\alpha ^{k}\neq 1$ ${\ displaystyle \ alpha ^ {k} \ neq 1}$ at $0<k<Q-1$ ${\ displaystyle 0 <k <Q-1}$ ) fields $\mathbb {F} _{Q},\;Q=q^{m}$ ${\ displaystyle \ mathbb {F} _ {Q}, \; Q = q ^ {m}}$ then the cyclotomic class $C=\{\alpha ,\;\alpha ^{q},\;\alpha ^{q^{2}},\;\ldots ,\;\alpha ^{q^{m-1}}\}$ ${\ displaystyle C = \ {\ alpha, \; \ alpha ^ {q}, \; \ alpha ^ {q ^ {2}}, \; \ ldots, \; \ alpha ^ {q ^ {m-1} } \}}$ over the field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ will have exactly $m$ ${\ displaystyle m}$ elements.

It should be noted that any element of the cyclotomic class can generate this and only this class, and, therefore, belong only to it.

Examples of cyclotomic classes

Example 1. Let $q=2$ ${\ displaystyle q = 2}$ , $Q=2^{3}=8$ ${\ displaystyle Q = 2 ^ {3} = 8}$ and $\alpha$ ${\ displaystyle \ alpha}$ - primitive field element $\mathbb {F} _{8}$ ${\ displaystyle \ mathbb {F} _ {8}}$ , i.e $\alpha ^{7}=1$ ${\ displaystyle \ alpha ^ {7} = 1}$ and $\alpha ^{i}\neq 1$ ${\ displaystyle \ alpha ^ {i} \ neq 1}$ at $i<7$ ${\ displaystyle i <7}$ . Considering also that $\alpha ^{8}=\alpha$ ${\ displaystyle \ alpha ^ {8} = \ alpha}$ , we can obtain the expansion of all nonzero elements of the field $\mathbb {F} _{8}$ ${\ displaystyle \ mathbb {F} _ {8}}$ into three cyclotomic classes over the field $\mathbb {F} _{2}$ ${\ displaystyle \ mathbb {F} _ {2}}$ :

{\begin{matrix}\{1\},\\\{\alpha ,\;\alpha ^{2},\;\alpha ^{4}\},\\\{\alpha ^{3},\;\alpha ^{6},\;\alpha ^{5}\}.\end{matrix}}

{\ displaystyle {\ begin {matrix} \ {1 \}, \\\ {\ alpha, \; \ alpha ^ {2}, \; \ alpha ^ {4} \}, \\\ {\ alpha ^ { 3}, \; \ alpha ^ {6}, \; \ alpha ^ {5} \}. \ End {matrix}}}

Example 2. Similarly, we can construct classes on the field $\mathbb {F} _{16}$ ${\ displaystyle \ mathbb {F} _ {16}}$ over the field $\mathbb {F} _{4}$ ${\ displaystyle \ mathbb {F} _ {4}}$ , i.e $q=4,\;Q=q^{2}=16$ ${\ displaystyle q = 4, \; Q = q ^ {2} = 16}$ . Let be $\alpha$ ${\ displaystyle \ alpha}$ - primitive field element $\mathbb {F} _{16}$ ${\ displaystyle \ mathbb {F} _ {16}}$ means $\alpha ^{15}=1,\;\alpha ^{16}=\alpha$ ${\ displaystyle \ alpha ^ {15} = 1, \; \ alpha ^ {16} = \ alpha}$ .

{\begin{matrix}\{1\},\\\{\alpha ,\;\alpha ^{4}\},\;\{\alpha ^{2},\;\alpha ^{8}\},\\\{\alpha ^{3},\;\alpha ^{12}\},\;\{\alpha ^{5}\},\;\{\alpha ^{10}\},\\\{\alpha ^{6},\;\alpha ^{9}\},\;\{\alpha ^{7},\;\alpha ^{13}\},\\\{\alpha ^{11},\;\alpha ^{14}\}.\end{matrix}}

{\ displaystyle {\ begin {matrix} \ {1 \}, \\\ {\ alpha, \; \ alpha ^ {4} \}, \; \ {\ alpha ^ {2}, \; \ alpha ^ { 8} \}, \\\ {\ alpha ^ {3}, \; \ alpha ^ {12} \}, \; \ {\ alpha ^ {5} \}, \; \ {\ alpha ^ {10} \}, \\\ {\ alpha ^ {6}, \; \ alpha ^ {9} \}, \; \ {\ alpha ^ {7}, \; \ alpha ^ {13} \}, \\\ {\ alpha ^ {11}, \; \ alpha ^ {14} \}. \ end {matrix}}}

Connection with the roots of polynomials

The following theorem establishes a connection between cyclotomic classes and the expansion of a polynomial $x^{Q-1}-1$ ${\ displaystyle x ^ {Q-1} -1}$ to irreducible polynomials over a field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ .

Theorem 3. Let $\alpha ,\;\alpha ^{q},\;\alpha ^{q^{2}},\;\ldots ,\;\alpha ^{q^{m-1}}$ ${\ displaystyle \ alpha, \; \ alpha ^ {q}, \; \ alpha ^ {q ^ {2}}, \; \ ldots, \; \ alpha ^ {q ^ {m-1}}}$ element cyclotomic class $\alpha \in \mathbb {F} _{q^{l}}$ ${\ displaystyle \ alpha \ in \ mathbb {F} _ {q ^ {l}}}$ and polynomial $f(x)=f_{0}+f_{1}x+f_{2}x^{2}+\ldots +f_{m-1}x^{m-1}+x^{m}$ ${\ displaystyle f (x) = f_ {0} + f_ {1} x + f_ {2} x ^ {2} + \ ldots + f_ {m-1} x ^ {m-1} + x ^ {m }}$ has as its roots the elements of this cyclotomic class, i.e.

f(x)=(x-\alpha )(x-\alpha ^{q})\ldots (x-\alpha ^{q^{m-1}}).

{\ displaystyle f (x) = (x- \ alpha) (x- \ alpha ^ {q}) \ ldots (x- \ alpha ^ {q ^ {m-1}}).}

Then the coefficients $f_{0},\;f_{1},\;\ldots ,\;f_{m-1}$ ${\ displaystyle f_ {0}, \; f_ {1}, \; \ ldots, \; f_ {m-1}}$ polynomial $f(x)$ ${\ displaystyle f (x)}$ lie in the field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ , and the polynomial itself is irreducible over this field.

One can establish the following corollary from Theorem 3 . From the property of finite fields, which says that all nonzero elements of the field $\mathbb {F} _{Q}$ ${\ displaystyle \ mathbb {F} _ {Q}}$ are the roots of the polynomial $x^{Q-1}-1$ ${\ displaystyle x ^ {Q-1} -1}$ , we can conclude that the polynomial $x^{Q-1}-1$ ${\ displaystyle x ^ {Q-1} -1}$ can be decomposed into irreducible over the field $\mathbb {F} _{q}$ ${\ displaystyle \ mathbb {F} _ {q}}$ polynomials $f_{0}(x),\;f_{1}(x),\;\ldots ,\;f_{d}$ ${\ displaystyle f_ {0} (x), \; f_ {1} (x), \; \ ldots, \; f_ {d}}$ , each of which corresponds to its cyclotomic class ^[14] .

Types of Polynomials

Primitive polynomials

Definition The root order of the irreducible polynomial is called the exponent to which this polynomial belongs. An irreducible polynomial is called primitive if all its roots are generating elements of the multiplicative group of the field ^[15] .

All roots of a primitive polynomial have an order equal to the order of the multiplicative group of the extended field $\mathbb {F} _{Q}$ ${\ displaystyle \ mathbb {F} _ {Q}}$ , i.e $Q-1$ ${\ displaystyle Q-1}$ ^[11] .

Circular Polynomials

Let be $\alpha$ ${\ displaystyle \ alpha}$ is the generating element of the multiplicative group of the field $GF(q^{m})$ ${\ displaystyle GF (q ^ {m})}$ , and its order is $n=q^{m}-1$ ${\ displaystyle n = q ^ {m} -1}$ , i.e $\alpha ^{q^{m}-1}=1$ ${\ displaystyle \ alpha ^ {q ^ {m} -1} = 1}$ . Let all the elements of order $d$ ${\ displaystyle d}$ are the roots of the polynomial $\psi _{d}(x),\;n\,\vdots \,d$ ${\ displaystyle \ psi _ {d} (x), \; n \, \ vdots \, d}$ . Then such a polynomial is called circular and equality is true ^[16] :

x^{q^{m}-1}-1=\prod _{d\,\mid \,q^{m}-1}\psi _{d}(x)

{\ displaystyle x ^ {q ^ {m} -1} -1 = \ prod _ {d \, \ mid \, q ^ {m} -1} \ psi _ {d} (x)}

Zhegalkin polynomials

Among the polynomials over finite fields, Zhegalkin polynomials are especially distinguished. They are polynomials of many variables over a field $GF(2)$ ${\ displaystyle GF (2)}$ ^[17] .

f(x_{1},\;x_{2},\;\ldots ,\;x_{N})=a+\sum _{1\leqslant i\leqslant N}a_{i}x_{i}+\sum _{1\leqslant i<j\leqslant N}a_{i,j}x_{i}x_{j}+\sum _{1\leqslant i<j<k\leqslant N}a_{i,\;j,\;k}x_{i}x_{j}x_{k}+\ldots +a_{1,\;2,\;\ldots ,\;N}x_{1}x_{2}\ldots x_{N}

{\ displaystyle f (x_ {1}, \; x_ {2}, \; \ ldots, \; x_ {N}) = a + \ sum _ {1 \ leqslant i \ leqslant N} a_ {i} x_ {i } + \ sum _ {1 \ leqslant i <j \ leqslant N} a_ {i, j} x_ {i} x_ {j} + \ sum _ {1 \ leqslant i <j <k \ leqslant N} a_ {i , \; j, \; k} x_ {i} x_ {j} x_ {k} + \ ldots + a_ {1, \; 2, \; \ ldots, \; N} x_ {1} x_ {2} \ ldots x_ {N}}

With the help of such a polynomial, any Boolean function can be specified ^[18] $f(x_{1},\;x_{2},\;\ldots ,\;x_{N})$ ${\ displaystyle f (x_ {1}, \; x_ {2}, \; \ ldots, \; x_ {N})}$ , and uniquely ^[17] ^[19] .

Application

There are many algorithms that use polynomials over finite fields and rings.

Euclidean Algorithm
Burlekamp Algorithm
Burlekamp Algorithm - Massey
Burlekamp Method
Bowes Code - Chowdhury - Hockingham
Reed - Solomon Code
CRC
Agrawala Test - Kayala - Saxen
Shamir's secret sharing scheme

Also, polynomials over finite fields are used in modern noise-resistant coding ^[20] (for describing cyclic codes ^[21] and for decoding the Reed – Solomon code using the Euclidean algorithm ^[22] ), pseudorandom number generators ^[23] (implemented using shift registers ) ^[24] , stream encryption ^[25] and data integrity verification algorithms.

Notes

↑ Akritas, 1994 , p. 146.
↑ ¹ ² ³ Bleikhut, 1986 , p. 88.
↑ Bleichut, 1986 , p. 90.
↑ Bleichut, 1986 , p. 91.
↑ ¹ ² Bleikhut, 1986 , p. 89.
↑ ¹ ² Sagalovich, 2014 , p. 79.
↑ Sagalovich, 2014 , p. 93.
↑ Bleichut, 1986 , p. 104.
↑ ¹ ² Sagalovich, 2014 , p. 101.
↑ Sagalovich, 2014 , p. 82.
↑ ¹ ² Sagalovich, 2014 , p. 96.
↑ Sagalovich, 2014 , p. 105.
↑ Bleichut, 1986 , p. 99.
↑ Sagalovich, 2014 , p. 97.
↑ Bleichut, 1986 , p. 103.
↑ Sagalovich, 2014 , p. 102.
↑ ¹ ² Yablonsky, 1986 , p. 32.
↑ Yablonsky, 1986 , p. 12.
↑ Gabidulin, Kshevetsky, Kolybelnikov, 2011 , p. 81.
↑ Sagalovich, 2014 , p. 169-172.
↑ Bleichut, 1986 , p. 116-121.
↑ Bleichut, 1986 , p. 223-228.
↑ Gabidulin, Kshevetsky, Kolybelnikov, 2011 , p. 77-83.
↑ Alferov, Teeth, Kuzmin et al., 2005 , p. 251-260.
↑ Gabidulin, Kshevetsky, Kolybelnikov, 2011 , p. 74-83.

Literature

Akritas A. Fundamentals of computer algebra with applications / trans. from English E.V. Pankratieva . - M .: Mir, 1994 .-- 544 p.
Alferov A. P. , Zubov A. Yu. , Kuzmin A. S. et al. Fundamentals of cryptography : Textbook - 3rd ed., Rev. and add. - M .: Helios ARV , 2005 .-- 480 p. - ISBN 978-5-85438-137-6
<a href=" https://wikidata.org/wiki/Track:Q22304400 "> </a> <a href=" https://wikidata.org/wiki/Track:Q27449770 "> </a> <a href = " https://wikidata.org/wiki/Track:Q27048435 "> </a> <a href=" https://wikidata.org/wiki/Track:Q27976247 "> </a> <a href = " https://wikidata.org/wiki/Track:Q27449771 "> </a> <a href=" https://wikidata.org/wiki/Track:Q27449773 "> </a>
Bleikhut R. E. Theory and Practice of Error Control Codes / Ed. K. Sh. Zigangirova - M .: Mir , 1986 .-- 576 p.
<a href=" https://wikidata.org/wiki/Track:Q2440867 "> </a> <a href=" https://wikidata.org/wiki/Track:Q27976630 "> </a> <a href = " https://wikidata.org/wiki/Track:Q27976629 "> </a> <a href=" https://wikidata.org/wiki/Track:Q7324196 "> </a>
Gabidulin E.M. , Kshevetsky A.S. , Kolybelnikov A.I. Information security : a training manual - M .: MIPT , 2011. - 225 p. - ISBN 978-5-7417-0377-9
<a href=" https://wikidata.org/wiki/Track:Q4130888 "> </a> <a href=" https://wikidata.org/wiki/Track:Q26887236 "> </a> <a href = " https://wikidata.org/wiki/Track:Q26887244 "> </a> <a href=" https://wikidata.org/wiki/Track:Q26887242 "> </a>
Sagalovich Yu. L. Introduction to algebraic codes - 3rd ed., Rev. and add. - M .: IPPI RAS , 2014 .-- 310 p. - ISBN 978-5-901158-24-1
<a href=" https://wikidata.org/wiki/Track:Q21789632 "> </a> <a href=" https://wikidata.org/wiki/Track:Q21789631 "> </a>
Yablonsky S. V. Introduction to discrete mathematics : Textbook. manual for universities - 2nd ed., revised. and add. - M .: Nauka , 1986 .-- 384 p.
<a href=" https://wikidata.org/wiki/Track:Q248326 "> </a> <a href=" https://wikidata.org/wiki/Track:Q27976594 "> </a> <a href = " https://wikidata.org/wiki/Track:Q3710055 "> </a>
Peterson W., Weldon E. Error Correcting Codes. - M .: Mir, 1976 .-- S. 596.

[_194b344bce494665-1] Akritas, 1994 , p. 146.

[_753325bfb512e3fe-2] ¹ ² ³ Bleikhut, 1986 , p. 88.

[_753326bfb512e549-3] Bleichut, 1986 , p. 90.

[_753326bfb512e548-4] Bleichut, 1986 , p. 91.

[_753325bfb512e3ff-5] ¹ ² Bleikhut, 1986 , p. 89.

[_0969f51995f6bddb-6] ¹ ² Sagalovich, 2014 , p. 79.

[_0969f71995f6c17b-7] Sagalovich, 2014 , p. 93.

[_38eb90c0af332127-8] Bleichut, 1986 , p. 104.

[_f5b4c879d23302b1-9] ¹ ² Sagalovich, 2014 , p. 101.

[_0969f81995f6c2c9-10] Sagalovich, 2014 , p. 82.

[_0969f71995f6c17e-11] ¹ ² Sagalovich, 2014 , p. 96.

[_f5b4c879d23302b5-12] Sagalovich, 2014 , p. 105.

[_753326bfb512e540-13] Bleichut, 1986 , p. 99.

[_0969f71995f6c17f-14] Sagalovich, 2014 , p. 97.

[_38eb90c0af332120-15] Bleichut, 1986 , p. 103.

[_f5b4c879d23302b2-16] Sagalovich, 2014 , p. 102.

[_983607084ebbf43b-17] ¹ ² Yablonsky, 1986 , p. 32.

[_983605084ebbf091-18] Yablonsky, 1986 , p. 12.

[_41573a8174d4d05f-19] Gabidulin, Kshevetsky, Kolybelnikov, 2011 , p. 81.

[_8ca4373fce3d958b-20] Sagalovich, 2014 , p. 169-172.

[_db78abbef20e0dee-21] Bleichut, 1986 , p. 116-121.

[_071c8385b9e130b3-22] Bleichut, 1986 , p. 223-228.

[_cb800f78128d77e5-23] Gabidulin, Kshevetsky, Kolybelnikov, 2011 , p. 77-83.

[_fb59ab1e2aa6539c-24] Alferov, Teeth, Kuzmin et al., 2005 , p. 251-260.

[_9c5cd6e6e50b742c-25] Gabidulin, Kshevetsky, Kolybelnikov, 2011 , p. 74-83.