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Gilbert-Varshamov Bound

Gilbert-Varshamov bound

The Gilbert-Varshamov bound is a bound on the parameters of a (not necessarily linear) code. It is occasionally known as the Gilbert-Shannon-Varshamov bound (or the GSV bound), but the Gilbert-Varshamov bound is by far the most popular name.

Statement of the bound

Let :

A_q(n,d)

denote the maximum possible size of a q-ary code

C

with length

n

and minimum Hamming weight

d

(a q-ary code is a linear code over the field

\mathbb_q^n

q

elements). Then: :

\begin
\\
A_q(n,d) \geq \frac \\
\\
\end

Proof

Let

C

be a code of length

n

and minimum Hamming distance

d

having maximal size: :

|C|=A_q(n,d)

. Then for all

x\in\mathbb_q^n

there exists a codeword

c \in C

such that the Hamming distance

d(x,c)

between

x

and

c

satisfies :

d(x,c)\leq d-1

since otherwise we could add

x

to the code whilst maintaining the code's minimum Hamming distance

d

- a contradiction on the maximality of

|C|

. Hence the whole of

\mathbb_q^n

is contained in the union of all balls of radius

d-1

having their centre at some

c \in C

: :

\mathbb_q^n \subset \cup_ B(c,d-1)

. Now each ball has size :

\begin
\sum_^ \binom(q-1)^k \\
\end

since we may allow (or choose) upto

d-1

of the

n

components of a codeword to deviate (from the value of the corresponding component of the ball's centre) to one of

(q-1)

possible other values (recall: the code is q-ary: it takes values in

\mathbb_q^n

). Hence we deduce :

\begin
|\mathbb_q^n| & \leq & |\cup_ B(c,d-1)| \\
\\
& = & |C|\sum_^ \binom(q-1)^k \\
\\
\end

That is: :

\begin
A(n,d) & \geq & \frac \\
\end

(using the fact:

|\mathbb_q^n|=q^n

Linear code

In mathematics, a (binary) linear code of length

1 \leq n

and rank

1\leq k \leq n

is a linear subspace

C

with dimension

k

of the vector space :

\mathbb^n_2

. Aside:

\mathbb_2 = \

is the field of two elements and

\mathbb^n_2

is the set of all n-tuples of length

n

over

\mathbb_2

. Occasionally some other finite field

\mathbb_q

containing

q > 2

elements is used, in which case the code is said to be a "q-ary" code (rather than a binary code). Special exceptions to the general adjective "q-ary" are binary and ternary codes (corresponding to q=2 and q=3 respectively).

Properties

By virtue of the fact that the code is a subspace of

\mathbb^n_2

, the sum

c_1 + c_2

of two codewords in

C

is also a codeword (ie an element of the subspace

C

). Thus the entire code (which may be very large) to be represented as the span of a minimal set of codewords (known as a basis in linear algebra terms). These basis codewords are often collated in the rows of a matrix known as a generating matrix for the code

C

. The subspace definition also gives rise to the important property that the minimum Hamming distance between codewords is simply the minimum Hamming weight of all codewords since: :

d(c_1, c_2)=d(c_1+c_2, 0)\,

implying: :

\min_d(c_1,c_2)=\min_d(c_1+c_2, 0)=\min_d(c_1, 0).

The motivation behind creating linear codes is to allow for syndrome decoding.

Popular notation

Codes in general are often denoted by the letter

C

. A linear code of length

n

, rank

k

(ie having

k

codewords in its basis and

k

rows in its generating matrix) and minimum Hamming weight

d

is referred to as an

(n,k,d)

code. Remark. This is not to be confused with the notation

[n,r,d]

to denote a non-linear code of length

n

, size

r

(ie having

r

codewords) and minimum Hamming distance

d

Field (mathematics)

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.

Introduction

Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of associativity, commutativity and distributivity hold. Fields also appear in many other areas of mathematics; see the examples below. When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a commutative field or a rational domain. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except for commutativity, is today called a division ring or sometimes a skew field, but also non-commutative field is still widely used. Other languages have retained the old usage: for example, in Italian and French, division rings are called corpo and corps, both literally meaning 'body'. Instead, in German and Spanish, Körper (whence the blackboard bold K used to denote a field) and cuerpo mean 'field'. Notice that French language has no single word for field, they are simply called corps commutatif. Italian for field is campo, with the same literal meaning as English. The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field. Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other. See field theory for more information.

Definition

A field is a commutative ring (F, +,
- ) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. Spelled out, this means that the following hold: ; Closure of F under + and
- : For all a, b belonging to F, both a + b and a
- b belong to F (or more formally, + and
- are binary operations on F). ; Both + and
- are associative : For all a, b, c in F, a + (b + c) = (a + b) + c and a
- (b
- c) = (a
- b)
- c. ; Both + and
- are commutative : For all a, b belonging to F, a + b = b + a and a
- b = b
- a. ; The operation
- is distributive over the operation + : For all a, b, c, belonging to F, a
- (b + c) = (a
- b) + (a
- c). ; Existence of an additive identity : There exists an element 0 in F, such that for all a belonging to F, a + 0 = a. ; Existence of a multiplicative identity : There exists an element 1 in F different from 0, such that for all a belonging to F, a
- 1 = a. ; Existence of additive inverses : For every a belonging to F, there exists an element −a in F, such that a + (−a) = 0. ; Existence of multiplicative inverses : For every a ≠ 0 belonging to F, there exists an element a⁻¹ in F, such that a
- a⁻¹ = 1. The requirement 0 ≠ 1 ensures that the set which only contains a single element is not a field. Directly from the axioms, one may show that (F, +) and (F − ,
- ) are commutative groups (abelian groups) and that therefore (see elementary group theory) the additive inverse −a and the multiplicative inverse a⁻¹ are uniquely determined by a. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses: :(a
- b)⁻¹ = b⁻¹
- a⁻¹ = a⁻¹
- b⁻¹ provided both a and b are non-zero. Other useful rules include :−a = (−1)
- a and more generally :−(a
- b) = (−a)
- b = a
- (−b) as well as :a
- 0 = 0, all rules familiar from elementary arithmetic. If the requirement of commutativity of the operation
- is dropped, one distinguishes the above commutative fields from non-commutative fields, usually called division rings or skew fields).

Examples of fields

- The complex numbers C, under the usual operations of addition and multiplication. The field of complex numbers contains the following subfields (a subfield of a field F is a set containing 0 and 1, closed under the operations + and
- of F and with its own operations defined by restriction):
- The rational numbers Q = where Z is the set of integers. The rational number field contains no proper subfields.
- An algebraic number field is a finite field extension of the rational numbers Q, that is, a field containing Q which has finite dimension as a vector space over Q. Such fields are very important in number theory.
- The field of algebraic numbers, the algebraic closure of Q.
- The real numbers R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered field which is categorical — it is this structure that provides the foundation for most formal treatments of calculus.
- The real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers, and the definable numbers.
- If q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements, usually denoted F_q, Z/qZ, or GF(q). Every other finite field is isomorphic to one of these fields. Such fields are often called a Galois field, whence the notation GF(q).
- In particular, for a given prime number p, the set of integers modulo p is a finite field with p elements: F_p = where the operations are defined by performing the operation in Z, dividing by p and taking the remainder; see modular arithmetic.
- Taking p = 2, we obtain the smallest field, F₂, which has only two elements: 0 and 1. It can be defined by the two Cayley tables + 0 1
- 0 1 0 0 1 0 0 0 1 1 0 1 0 1 ::::This field has important uses in computer science, especially in cryptography and coding theory.
- The rational numbers can be extended to the fields of p-adic numbers for every prime number p. These fields are very important in both number theory and mathematical analysis.
- Let E and F be two fields with E a subfield of F. Let x be an element of F not in E. Then E(x) is defined to be the smallest subfield of F containing E and x. We call E(x) a simple extension of E. For instance, Q(i) is the number field of complex numbers C consisting of all numbers of the form a + bi where both a and b are rational numbers. In fact, it can be shown that every number field is a simple extension of Q.
- For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F. This is the simplest example of a transcendental extension.
- If F is a field, and p(X) is an irreducible polynomial in the polynomial ring F[X], then the quotient F[X]/<p(X)> is a field with a subfield isomorphic to F. For instance, R[X]/<X² + 1> is a field (in fact, it is isomorphic to the field of complex numbers). It can be shown that every simple algebraic extension of F is isomorphic to a field of this form.
- When F is a field, the set F((X)) of formal Laurent series over F is a field.
- If V is an algebraic variety over F, then the rational functions V → F form a field, the function field of V.
- If S is a Riemann surface, then the meromorphic functions S → C form a field.
- If I is an index set, U is an ultrafilter on I, and F_i is a field for every i in I, the ultraproduct of the F_i (using U) is a field.
- Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers. There are also proper classes with field structure, which are sometimes called Fields, with a capital F:
- The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The set of all surreal numbers with birthday smaller than some inaccessible cardinal form a field.
- The nimbers form a Field. The set of nimbers with birthday smaller than

2^

, the nimbers with birthday smaller than any infinite cardinal are all examples of fields.

Some first theorems

- The set of non-zero elements of a field F (typically denoted by F^×) is an abelian group under multiplication. Every finite subgroup of F^× is cyclic.
- The characteristic of any field is zero or a prime number. (The characteristic is defined as follows: the smallest positive integer n such that n·1 = 0, or zero if no such n exists; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. An equivalent definition is the following: the characteristic of a field F is the unique non-negative generator of the kernel of the unique ring homomorphism Z → F which sends 1 |-> 1.)
- The number of elements of any finite field is a prime power.
- As a ring, a field has no ideals except and itself.
- Assuming the axiom of choice, for every field F, there exists a unique field G (up to isomorphism) which contains F, is algebraic over F, and is algebraically closed. G is called the algebraic closure of F.

Union (set theory)

In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. This article uses mathematical symbols.

Basic definition

mathematical symbols If A and B are sets, then the union of A and B is the set that contains all elements of A and all elements of B, but no other elements. The union of A and B is usually written "A ∪ B". Formally: : x is an element of A ∪ B if and only if :
- x is an element of A or :
- x is an element of B. (This is an inclusive "or".) For example, the union of the sets and is . The number 9 is not contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even.

Finite unions

More generally, one can take the union of several sets at once. The union of A, B, and C, for example, contains all elements of A, all elements of B, and all elements of C, and nothing else. Formally, x is an element of A ∪ B ∪ C if x is in A or x is in B or x is in C. Union is an associative operation, it doesn't matter in what order unions are taken. In mathematics a finite union means any union carried out on a finite number of sets: it doesn't imply that the union set is a finite set.

Algebraic properties

Binary union (the union of just two sets at a time) is an associative operation; that is, :A ∪(B ∪ C) = (A ∪ B) ∪ C. In fact, A ∪ B ∪ C is equal to both of these sets as well, so parentheses are never needed when writing only unions. Similarly, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union. That is, ∪ A = A, for any set A. Thus one can think of the empty set as the union of zero sets. In terms of the definitions, these facts follow from analogous facts about logical disjunction. Together with intersection and complement, union makes any power set into a Boolean algebra. For example, union and intersection distribute over each other, and all three operations are combined in De Morgan's laws. Replacing union with symmetric difference gives a Boolean ring instead of a Boolean algebra.

Infinite unions

The most general notion is the union of an arbitrary collection of sets. If M is a set whose elements are themselves sets, then x is an element of the union of M if and only if for at least one element A of M, x is an element of A. In symbols: :

x \in \bigcup\mathbf \iff \exists A\mathbf, x \in A.

That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union in axiomatic set theory. This idea subsumes the above paragraphs, in that for example, A ∪ B ∪ C is the union of the collection . Also, if M is the empty collection, then the union of M is the empty set. The analogy between finite unions and logical disjunction extends to one between infinite unions and existential quantification. The notation for the general concept can vary considerably. Hardcore set theorists will simply write :

\bigcup \mathbf,

while most people will instead write :

\bigcup_ A.

The latter notation can be generalised to :

\bigcup_ A_,

which refers to the union of the collection . Here I is a set, and A_i is a set for every i in I. In the case that the index set I is the set of natural numbers, the notation is analogous to that of infinite series: :

\bigcup_^ A_.

When formatting is difficult, this can also be written "A₁ ∪ A₂ ∪ A₃ ∪ ···". (This last example, a union of countably many sets, is very common in analysis; for an example see the article on σ-algebras.) Finally, let us note that whenever the symbol "∪" is placed before other symbols instead of between them, it is of a larger size. Intersection distributes over infinitary union, in the sense that :

\bigcup_ (A \cap B_) = A \cap \bigcup_ B_.

We can also combine infinitary union with infinitary intersection to get the law :

\bigcup_ (\bigcap_ A_) \subseteq \bigcap_ (\bigcup_ A_).

Ball (mathematics)

In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general.

Metric spaces

Let M be a metric space. The (open) ball of radius r > 0 centred at a point p in M is defined as :

B_r(p) = \,

where d is the distance function or metric. If the less-than symbol (<) is replaced by a less-than-or-equal-to (≤), the above definition becomes that of a closed ball: :

_r(p) = \

. Note in particular that a ball (open or closed) always includes p itself, since r > 0. A (open or closed) unit ball is a ball of radius 1. In n-dimensional Euclidean space with the ordinary (Euclidean) metric, if the space is the line, the ball is an interval, and if the space is the plane, the ball is the disc inside a circle. A closed unit ball is denoted by Dⁿ; its outside is the n-1-sphere S^n-1, e.g. the 3-sphere S³ is the outside of D⁴ in 4D. These and the corresponding objects in even higher dimensions are called hyperball and hypersphere. See the latter for "volumes" and "areas". With other metrics the shape of a ball can be different; examples:
- in 2D:
- with the 1-norm (i.e. in taxicab geometry) a ball is a square with the diagonals parallel to the coordinate axes
- with the Chebyshev distance a ball is a square with the sides parallel to the coordinate axes
- in 3D:
- with the 1-norm a ball is a regular octahedron with the body diagonals parallel to the coordinate axes
- with the Chebyshev distance a ball is a cube with the edges parallel to the coordinate axes Note that in most cases rotated balls are not balls.

Related notions

Open balls with respect to a metric d form a basis for the topology induced by d (by definition). This means, among other things, that all open sets in a metric space can be written as a union of open balls. A subset of a metric space is bounded if it is contained in a ball. A set is totally bounded if given any radius, it is covered by finitely many balls of that radius.

Binomial coefficients

:See binomial (disambiguation) for a list of other topics called by that name. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number :

= \frac = \frac \quad \mbox n\geq k\geq 0 \qquad \mbox

and :

= 0 \quad \mbox k<0 \mbox k>n

where m! denotes the factorial of m. According to Nicholas J. Higham, the :

notation was introduced by Albert von Ettinghausen in 1826, although these numbers have been known centuries before that; see Pascal's triangle. The binomial coefficient of n and k is also written as C(n, k), _nC_k or

C^_

(C for combination) and read as "n choose k". For compactness, from here on we will use the first of these three notations. The binomial coefficients are the coefficients in the expansion of the binomial (x + y)ⁿ (hence the name): :

(x+y)^n = \sum_^ C(n, k) x^ y^k. \qquad (2)

This is generalized by the binomial theorem, which allows the exponent n to be negative or a non-integer. The importance of the binomial coefficients lies in the fact that C(n, k) is the number of ways that k objects can be chosen from n objects, regardless of order. See the article on combination.

Example

\mathrm(7, 3) = \frac = \frac = 35.

The computation is conveniently arranged like this :

\mathrm(7,3) = ((((5/1)6)/2)7)/3 = ((30/2)7)/3 = 105/3 = 35.

Derivation from binomial expansion

For exponent 1, (x+y)¹ is x+y. For exponent 2, (x+y)² is (x+y)(x+y), which forms terms as follows. The first factor supplies either an x or a y; likewise for the second factor. Thus to form x², the only possibility is to choose x from both factors; likewise for y². However, the xy term can be formed by x from the first and y from the second factor, or y from the first and x from the second factor; thus it acquires a coefficient of 2. Proceeding to exponent 3, (x+y)³ reduces to (x+y)²(x+y), where we already know that (x+y)² = x²+2xy+y². Again the extremes, x³ and y³ arise in a unique way. However, the term x²y is either 2xy times x or x² times y, for a coefficient of 3; likewise xy² arises in two ways, summing the coefficients 1 and 2 to give 3. This suggests an induction. Thus for exponent 4, each term has total degree (sum of exponents) 4 (in general, n), with 4−k factors of x and k factors of y. If k is neither 0 nor 1 (terms x⁴ or y⁴), then the term arises in two ways, from adjacent coefficients with total degree 3. For example, x²y² is both xy² times x and x²y times y, thus its coefficient is 3+3. This is the origin of Pascal's triangle, discussed below. Another perspective is that to form x^n−ky^k from n factors of (x+y), we must choose y from k of the factors and x from the rest. To count the possibilities, consider all n! permutations of the factors. Represent each permutation as a shuffled list of the numbers from 1 to n. Select an x from the first n−k factors listed, and a y from the remaining k factors; in this way each permutation contributes to the term x^n−ky^k. For example, the list ⟨4,1,2,3⟩ selects x from factors 4 and 1, and selects y from factors 2 and 3, as one way to form the term x²y². : (x +¹ y) (x +² y) (x +³ y) (x +⁴ y) But the distinct list ⟨1,4,3,2⟩ makes exactly the same selection; the binomial coefficient formula must remove this redundancy. The n−k factors for x have (n−k)! permutations, and the k factors for y have k! permutations. Therefore n! / (n−k)! k! is the number of truly distinct ways to form the term x^n−ky^k. This discussion extends to the case when each factor is a sum of multiple variables, leading naturally to the definition of a multinomial coefficient. A convenient notation uses a list of variables, x = (x₁,…,x_m), with the exponents given as another list, E = (e₁,…,e_m), called a multi-index. The terms in the expansion of (x₁+⋯+x_m)ⁿ have the form :

^E = x_1^ x_2^ \cdots x_m^,

where |E| = e₁+⋯+e_m = n, and the coefficient of such a term is the multinomial coefficient :

\frac.

The simple binomial coefficients are the case m = 2.

Pascal's triangle

Pascal's rule is the important recurrence relation :

\mathrm(n,k) +  \mathrm(n,k+1) = C(n+1,k+1), \qquad (3)

which follows directly from the definition. This recurrence relation can be used to prove by mathematical induction that C(n, k) is a natural number for all n and k, a fact that is not immediately obvious from the definition. It also gives rise to Pascal's triangle: row 0 1 row 1 1 1 row 2 1 2 1 row 3 1 3 3 1 row 4 1 4 6 4 1 row 5 1 5 10 10 5 1 row 6 1 6 15 20 15 6 1 row 7 1 7 21 35 35 21 7 1 row 8 1 8 28 56 70 56 28 8 1 Row number n contains the numbers C(n, k) for k = 0,...,n. It is constructed by starting with ones at the outside and then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that :(x + y)⁵ = 1x⁵ + 5 x⁴y + 10 x³y² + 10 x²y³ + 5 x y⁴ + 1y⁵. The differences between elements on other diagonals are the elements in the previous diagonal - consequential to the recurrence relation (3) above. In the 1303 AD treatise Precious Mirror of the Four Elements, Zhu Shijie mentioned the triangle as an ancient method for solving binomial coefficients indicating that the method was known to Chinese mathematicians five centuries before Pascal.

Combinatorics and statistics

Binomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems:
- Every set with n elements has

\mathrm(n, k)

different subsets having k elements each (these are called k-combinations).
- The number of strings of length n containing k ones and n − k zeros is

\mathrm(n, k).

- There are

\mathrm(n+1, k)

strings consisting of k ones and n zeros such that no two ones are adjacent.
- The number of sequences consisting of n natural numbers whose sum equals k is

\mathrm(n+k-1, k)

; this is also the number of ways to choose k elements from a set of n if repetitions are allowed.
- The Catalan numbers have an easy formula involving binomial coefficients; they can be used to count various structures, such as trees and parenthesized expressions. The binomial coefficients also occur in the formula for the binomial distribution in statistics and in the formula for a Bézier curve.

Formulas involving binomial coefficients

One has that :

\mathrm(n,k)= \mathrm(n, n-k),\qquad\qquad(4)\,

which follows from expansion (2) by using (x + y)ⁿ = (y + x)ⁿ, and is reflected in the numerical "symmetry" of Pascal's triangle. Another formula is :

\sum_^ \mathrm(n,k) = 2^n; \qquad (5)

it is obtained from expansion (2) using x = y = 1. This is equivalent to saying that the elements in one row of Pascal's triangle always add up to two raised to an integer power. The formula :

\sum_^ k \mathrm(n,k) = n 2^ \qquad (6)

follows from expansion (2), after differentiating and substituting x = y = 1. Vandermonde's identity :

\sum_ \mathrm(m,j) \mathrm(n-m,k-j) = \mathrm(n,k) \qquad (7a)

is found by expanding (1+x)^m (1+x)^n-m = (1+x)ⁿ with (2). As C(n, k) is zero if k > n, the sum is finite for integer n and m. Equation (7a) generalizes equation (3). It holds for arbitrary, complex-valued

m

and

n

, the Chu-Vandermonde identity. A related formula is :

\sum_ \mathrm(m,j) \mathrm(n-m,k-j) = \mathrm(n+1,k+1). \qquad (7b)

While equation (7a) is true for all values of m, equation (7b) is true for all values of j. From expansion (7a) using n=2m, k = n, and (4), one finds :

\sum_^ \mathrm(m,j)^2 = \mathrm(2m,m). \qquad (8)

Denote by F(n + 1) the Fibonacci numbers. We obtain a formula about the diagonals of Pascal's triangle :

\sum_^ \mathrm(n-k,k) = \mathrm(n+1). \qquad (9)

This can be proved by induction using (3). Also using (3) and induction, one can show that :

\sum_^ \mathrm(j,k) = \mathrm(n+1,k+1). \qquad (10)

Divisors of binomial coefficients

The prime divisors of C(n, k) can be interpreted as follows: if p is a prime number and p^r is the highest power of p which divides C(n, k), then r is equal to the number of natural numbers j such that the fractional part of k/p^j is bigger than the fractional part of n/p^j. In particular, C(n, k) is always divisible by n/gcd(n,k).

Bounds for binomial coefficients

The following bounds for C(n, k) hold:
-

\mathrm(n, k)  \le \frac

\mathrm(n, k)  \le \left(\frac\right)^k

\mathrm(n, k)  \ge \left(\frac\right)^k

Generalization to real and complex argument

The binomial coefficient

can be defined for any complex number z and any natural number k as follows: :

= \prod_^= \frac \qquad (11)

This generalization is known as the generalized binomial coefficient and is used in the formulation of the binomial theorem and satisfies properties (3) and (7). For fixed k, the expression

f(z)=

is a polynomial in z of degree k with rational coefficients. f(z) is the unique polynomial of degree k satisfying :f(0) = f(1) = ... = f(k − 1) = 0 and :f(k) = 1. Any polynomial p(z) of degree d can be written in the form :

p(z) = \sum_^ a_k

This is important in the theory of difference equations and can be seen as a discrete analog of Taylor's theorem. Newton's binomial series gets the simple form :

(1+z)^ = \sum_^z^r = 1+z+z^2+\cdots.

It is not hard to show that the radius of convergence of this series is 1.

Newton's binomial series

Any polynomial p(z) of degree d can be written in the form :

p(z) = \sum_^ a_k

This is important in the theory of difference equations and can be seen as a discrete analog of Taylor's theorem. Newton's binomial series gets the simple form :

(1+z)^ = \sum_^z^r = 1+z+z^2+\cdots.

It is not hard to show that the radius of convergence of this series is 1.

Generalization to q-series

The binomial coefficient has a q-analog generalization known as the Gaussian binomial.

References

- This article incorporates material from the following PlanetMath articles, which are licensed under the GFDL: [http://planetmath.org/?op=getobj&from=objects&id=273 Binomial Coefficient], [http://planetmath.org/?op=getobj&from=objects&id=4074 Bounds for binomial coefficients], [http://planetmath.org/?op=getobj&from=objects&id=6744 Proof that C(n,k) is an integer], [http://planetmath.org/?op=getobj&from=objects&id=6309 Generalized binomial coefficients].
- Donald Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89683-4. Section 1.2.6: Binomial Coefficients, pp.52–74. Category:Combinatorics Category:Number sequences ko:이항계수

Singleton bound

The singleton bound is a relatively crude bound on the size of a code

C

of length

n

, size

r

and minimum Hamming weight

d

. Let :

A_q(n,d)

denote the maximum possible size of a q-ary code with length

n

and minimum Hamming weight

d

(a q-ary code is a linear code over the field of

q

elements). Then: :

A_q(n,d) \leq q^

Proof

First observe that the maximum size

r

of a q-ary code of length

n

q^n

since each component of a given codeword may take one of

q

different values independently of all other components. Let

C

be a q-ary code. Then clearly all codewords

c \in C

are distinct. If we delete the first

d-1

components of each codeword, then each resulting codeword must still be distinct since all codewords have Hamming weight at least

d

from each other. Thus the size

r

of the code is unchanged. The new code has length :

n-(d-1)=n-d+1

and thus has maximum possible size :

q^

(by the argument presented in the first paragraph of the proof). Hence the original code shares the same bound on its size: :

A_q(n,d) \leq q^

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(Kesen ku di vê rojê de ji dayîk bûn)

Mirin

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Gilbert-Varshamov bound

Statement of the bound

Proof

See also

Linear code

Properties

Popular notation

See also

Field (mathematics)

Introduction

Definition

Examples of fields

Some first theorems

See also

Union (set theory)

Basic definition

Finite unions

Algebraic properties

Infinite unions

See also

Ball (mathematics)

Metric spaces

Related notions

See also

Binomial coefficients

Example

Derivation from binomial expansion

Pascal's triangle

Combinatorics and statistics

Formulas involving binomial coefficients

Divisors of binomial coefficients

Bounds for binomial coefficients

Generalization to real and complex argument

Newton's binomial series

Generalization to q-series

See also

References

Singleton bound

Proof

See also

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Mirin

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