# Bounds for Parameters of Codes¶

This module provided some upper and lower bounds for the parameters of codes.

AUTHORS:

• David Joyner (2006-07): initial implementation.
• William Stein (2006-07): minor editing of docs and code (fixed bug in elias_bound_asymp)
• David Joyner (2006-07): fixed dimension_upper_bound to return an integer, added example to elias_bound_asymp.
• ” (2009-05): removed all calls to Guava but left it as an option.
• Dima Pasechnik (2012-10): added LP bounds.

Let $$F$$ be a finite field (we denote the finite field with $$q$$ elements by $$\GF{q}$$). A subset $$C$$ of $$V=F^n$$ is called a code of length $$n$$. A subspace of $$V$$ (with the standard basis) is called a linear code of length $$n$$. If its dimension is denoted $$k$$ then we typically store a basis of $$C$$ as a $$k\times n$$ matrix (the rows are the basis vectors). If $$F=\GF{2}$$ then $$C$$ is called a binary code. If $$F$$ has $$q$$ elements then $$C$$ is called a $$q$$-ary code. The elements of a code $$C$$ are called codewords. The information rate of $$C$$ is

$R={\frac{\log_q\vert C\vert}{n}},$

where $$\vert C\vert$$ denotes the number of elements of $$C$$. If $${\bf v}=(v_1,v_2,...,v_n)$$, $${\bf w}=(w_1,w_2,...,w_n)$$ are vectors in $$V=F^n$$ then we define

$d({\bf v},{\bf w}) =\vert\{i\ \vert\ 1\leq i\leq n,\ v_i\not= w_i\}\vert$

to be the Hamming distance between $${\bf v}$$ and $${\bf w}$$. The function $$d:V\times V\rightarrow \Bold{N}$$ is called the Hamming metric. The weight of a vector (in the Hamming metric) is $$d({\bf v},{\bf 0})$$. The minimum distance of a linear code is the smallest non-zero weight of a codeword in $$C$$. The relatively minimum distance is denoted

$\delta = d/n.$

A linear code with length $$n$$, dimension $$k$$, and minimum distance $$d$$ is called an $$[n,k,d]_q$$-code and $$n,k,d$$ are called its parameters. A (not necessarily linear) code $$C$$ with length $$n$$, size $$M=|C|$$, and minimum distance $$d$$ is called an $$(n,M,d)_q$$-code (using parentheses instead of square brackets). Of course, $$k=\log_q(M)$$ for linear codes.

What is the “best” code of a given length? Let $$F$$ be a finite field with $$q$$ elements. Let $$A_q(n,d)$$ denote the largest $$M$$ such that there exists a $$(n,M,d)$$ code in $$F^n$$. Let $$B_q(n,d)$$ (also denoted $$A^{lin}_q(n,d)$$) denote the largest $$k$$ such that there exists a $$[n,k,d]$$ code in $$F^n$$. (Of course, $$A_q(n,d)\geq B_q(n,d)$$.) Determining $$A_q(n,d)$$ and $$B_q(n,d)$$ is one of the main problems in the theory of error-correcting codes.

These quantities related to solving a generalization of the childhood game of “20 questions”.

GAME: Player 1 secretly chooses a number from $$1$$ to $$M$$ ($$M$$ is large but fixed). Player 2 asks a series of “yes/no questions” in an attempt to determine that number. Player 1 may lie at most $$e$$ times ($$e\geq 0$$ is fixed). What is the minimum number of “yes/no questions” Player 2 must ask to (always) be able to correctly determine the number Player 1 chose?

If feedback is not allowed (the only situation considered here), call this minimum number $$g(M,e)$$.

Lemma: For fixed $$e$$ and $$M$$, $$g(M,e)$$ is the smallest $$n$$ such that $$A_2(n,2e+1)\geq M$$.

Thus, solving the solving a generalization of the game of “20 questions” is equivalent to determining $$A_2(n,d)$$! Using Sage, you can determine the best known estimates for this number in 2 ways:

1. Indirectly, using best_known_linear_code_www(n, k, F),

which connects to the website http://www.codetables.de by Markus Grassl;

2. codesize_upper_bound(n,d,q), dimension_upper_bound(n,d,q),

and best_known_linear_code(n, k, F).

The output of best_known_linear_code(), best_known_linear_code_www(), or dimension_upper_bound() would give only special solutions to the GAME because the bounds are applicable to only linear codes. The output of codesize_upper_bound() would give the best possible solution, that may belong to a linear or nonlinear code.

This module implements:

• codesize_upper_bound(n,d,q), for the best known (as of May, 2006) upper bound A(n,d) for the size of a code of length n, minimum distance d over a field of size q.
• dimension_upper_bound(n,d,q), an upper bound $$B(n,d)=B_q(n,d)$$ for the dimension of a linear code of length n, minimum distance d over a field of size q.
• gilbert_lower_bound(n,q,d), a lower bound for number of elements in the largest code of min distance d in $$\GF{q}^n$$.
• gv_info_rate(n,delta,q), $$log_q(GLB)/n$$, where GLB is the Gilbert lower bound and delta = d/n.
• gv_bound_asymp(delta,q), asymptotic analog of Gilbert lower bound.
• plotkin_upper_bound(n,q,d)
• plotkin_bound_asymp(delta,q), asymptotic analog of Plotkin bound.
• griesmer_upper_bound(n,q,d)
• elias_upper_bound(n,q,d)
• elias_bound_asymp(delta,q), asymptotic analog of Elias bound.
• hamming_upper_bound(n,q,d)
• hamming_bound_asymp(delta,q), asymptotic analog of Hamming bound.
• singleton_upper_bound(n,q,d)
• singleton_bound_asymp(delta,q), asymptotic analog of Singleton bound.
• mrrw1_bound_asymp(delta,q), “first” asymptotic McEliese-Rumsey-Rodemich-Welsh bound for the information rate.
• Delsarte (a.k.a. Linear Programming (LP)) upper bounds.

PROBLEM: In this module we shall typically either (a) seek bounds on k, given n, d, q, (b) seek bounds on R, delta, q (assuming n is “infinity”).

TODO:

• Johnson bounds for binary codes.
• mrrw2_bound_asymp(delta,q), “second” asymptotic McEliese-Rumsey-Rodemich-Welsh bound for the information rate.

REFERENCES:

• C. Huffman, V. Pless, Fundamentals of error-correcting codes, Cambridge Univ. Press, 2003.
sage.coding.code_bounds.codesize_upper_bound(n, d, q, algorithm=None)

This computes the minimum value of the upper bound using the methods of Singleton, Hamming, Plotkin, and Elias.

If algorithm=”gap” then this returns the best known upper bound $$A(n,d)=A_q(n,d)$$ for the size of a code of length n, minimum distance d over a field of size q. The function first checks for trivial cases (like d=1 or n=d), and if the value is in the built-in table. Then it calculates the minimum value of the upper bound using the algorithms of Singleton, Hamming, Johnson, Plotkin and Elias. If the code is binary, $$A(n, 2\ell-1) = A(n+1,2\ell)$$, so the function takes the minimum of the values obtained from all algorithms for the parameters $$(n, 2\ell-1)$$ and $$(n+1, 2\ell)$$. This wraps GUAVA’s (i.e. GAP’s package Guava) UpperBound( n, d, q ).

If algorithm=”LP” then this returns the Delsarte (a.k.a. Linear Programming) upper bound.

EXAMPLES:

sage: codesize_upper_bound(10,3,2)
93
sage: codesize_upper_bound(24,8,2,algorithm="LP")
4096
sage: codesize_upper_bound(10,3,2,algorithm="gap")  # optional - gap_packages (Guava package)
85
sage: codesize_upper_bound(11,3,4,algorithm=None)
123361
sage: codesize_upper_bound(11,3,4,algorithm="gap")  # optional - gap_packages (Guava package)
123361
sage: codesize_upper_bound(11,3,4,algorithm="LP")
109226

sage.coding.code_bounds.dimension_upper_bound(n, d, q, algorithm=None)

Returns an upper bound $$B(n,d) = B_q(n,d)$$ for the dimension of a linear code of length n, minimum distance d over a field of size q. Parameter “algorithm” has the same meaning as in codesize_upper_bound()

EXAMPLES:

sage: dimension_upper_bound(10,3,2)
6
sage: dimension_upper_bound(30,15,4)
13
sage: dimension_upper_bound(30,15,4,algorithm="LP")
12

sage.coding.code_bounds.elias_bound_asymp(delta, q)

Computes the asymptotic Elias bound for the information rate, provided $$0 < \delta < 1-1/q$$.

EXAMPLES:

sage: elias_bound_asymp(1/4,2)
0.39912396330...

sage.coding.code_bounds.elias_upper_bound(n, q, d, algorithm=None)

Returns the Elias upper bound for number of elements in the largest code of minimum distance d in $$\GF{q}^n$$. Wraps GAP’s UpperBoundElias.

EXAMPLES:

sage: elias_upper_bound(10,2,3)
232
sage: elias_upper_bound(10,2,3,algorithm="gap")  # optional - gap_packages (Guava package)
232

sage.coding.code_bounds.entropy(x, q=2)

Computes the entropy at $$x$$ on the $$q$$-ary symmetric channel.

INPUT:

• x - real number in the interval $$[0, 1]$$.
• q - (default: 2) integer greater than 1. This is the base of the logarithm.

EXAMPLES:

sage: entropy(0, 2)
0
sage: entropy(1/5,4)
1/5*log(3)/log(4) - 4/5*log(4/5)/log(4) - 1/5*log(1/5)/log(4)
sage: entropy(1, 3)
log(2)/log(3)


Check that values not within the limits are properly handled:

sage: entropy(1.1, 2)
Traceback (most recent call last):
...
ValueError: The entropy function is defined only for x in the interval [0, 1]
sage: entropy(1, 1)
Traceback (most recent call last):
...
ValueError: The value q must be an integer greater than 1

sage.coding.code_bounds.entropy_inverse(x, q=2)

Find the inverse of the q-ary entropy function at the point x.

INPUT:

• x – real number in the interval $$[0, 1]$$.
• q - (default: 2) integer greater than 1. This is the base of the logarithm.

OUTPUT:

Real number in the interval $$[0, 1-1/q]$$. The function has multiple values if we include the entire interval $$[0, 1]$$; hence only the values in the above interval is returned.

EXAMPLES:

sage: from sage.coding.code_bounds import entropy_inverse
sage: entropy_inverse(0.1)
0.012986862055848683
sage: entropy_inverse(1)
1/2
sage: entropy_inverse(0, 3)
0
sage: entropy_inverse(1, 3)
2/3

sage.coding.code_bounds.gilbert_lower_bound(n, q, d)

Returns lower bound for number of elements in the largest code of minimum distance d in $$\GF{q}^n$$.

EXAMPLES:

sage: gilbert_lower_bound(10,2,3)
128/7

sage.coding.code_bounds.griesmer_upper_bound(n, q, d, algorithm=None)

Returns the Griesmer upper bound for number of elements in the largest code of minimum distance d in $$\GF{q}^n$$. Wraps GAP’s UpperBoundGriesmer.

EXAMPLES:

sage: griesmer_upper_bound(10,2,3)
128
sage: griesmer_upper_bound(10,2,3,algorithm="gap")  # optional - gap_packages (Guava package)
128

sage.coding.code_bounds.gv_bound_asymp(delta, q)

Computes the asymptotic GV bound for the information rate, R.

EXAMPLES:

sage: RDF(gv_bound_asymp(1/4,2))
0.18872187554086...
sage: f = lambda x: gv_bound_asymp(x,2)
sage: plot(f,0,1)
Graphics object consisting of 1 graphics primitive

sage.coding.code_bounds.gv_info_rate(n, delta, q)

GV lower bound for information rate of a q-ary code of length n minimum distance delta*n

EXAMPLES:

sage: RDF(gv_info_rate(100,1/4,3))
0.36704992608261894

sage.coding.code_bounds.hamming_bound_asymp(delta, q)

Computes the asymptotic Hamming bound for the information rate.

EXAMPLES:

sage: RDF(hamming_bound_asymp(1/4,2))
0.456435556800...
sage: f = lambda x: hamming_bound_asymp(x,2)
sage: plot(f,0,1)
Graphics object consisting of 1 graphics primitive

sage.coding.code_bounds.hamming_upper_bound(n, q, d)

Returns the Hamming upper bound for number of elements in the largest code of minimum distance d in $$\GF{q}^n$$. Wraps GAP’s UpperBoundHamming.

The Hamming bound (also known as the sphere packing bound) returns an upper bound on the size of a code of length n, minimum distance d, over a field of size q. The Hamming bound is obtained by dividing the contents of the entire space $$\GF{q}^n$$ by the contents of a ball with radius floor((d-1)/2). As all these balls are disjoint, they can never contain more than the whole vector space.

$M \leq {q^n \over V(n,e)},$

where M is the maximum number of codewords and $$V(n,e)$$ is equal to the contents of a ball of radius e. This bound is useful for small values of d. Codes for which equality holds are called perfect.

EXAMPLES:

sage: hamming_upper_bound(10,2,3)
93

sage.coding.code_bounds.mrrw1_bound_asymp(delta, q)

Computes the first asymptotic McEliese-Rumsey-Rodemich-Welsh bound for the information rate, provided $$0 < \delta < 1-1/q$$.

EXAMPLES:

sage: mrrw1_bound_asymp(1/4,2)
0.3545789026652697

sage.coding.code_bounds.plotkin_bound_asymp(delta, q)

Computes the asymptotic Plotkin bound for the information rate, provided $$0 < \delta < 1-1/q$$.

EXAMPLES:

sage: plotkin_bound_asymp(1/4,2)
1/2

sage.coding.code_bounds.plotkin_upper_bound(n, q, d, algorithm=None)

Returns Plotkin upper bound for number of elements in the largest code of minimum distance d in $$\GF{q}^n$$.

The algorithm=”gap” option wraps Guava’s UpperBoundPlotkin.

EXAMPLES:

sage: plotkin_upper_bound(10,2,3)
192
sage: plotkin_upper_bound(10,2,3,algorithm="gap")  # optional - gap_packages (Guava package)
192

sage.coding.code_bounds.singleton_bound_asymp(delta, q)

Computes the asymptotic Singleton bound for the information rate.

EXAMPLES:

sage: singleton_bound_asymp(1/4,2)
3/4
sage: f = lambda x: singleton_bound_asymp(x,2)
sage: plot(f,0,1)
Graphics object consisting of 1 graphics primitive

sage.coding.code_bounds.singleton_upper_bound(n, q, d)

Returns the Singleton upper bound for number of elements in the largest code of minimum distance d in $$\GF{q}^n$$. Wraps GAP’s UpperBoundSingleton.

This bound is based on the shortening of codes. By shortening an $$(n, M, d)$$ code d-1 times, an $$(n-d+1,M,1)$$ code results, with $$M \leq q^n-d+1$$. Thus

$M \leq q^{n-d+1}.$

Codes that meet this bound are called maximum distance separable (MDS).

EXAMPLES:

sage: singleton_upper_bound(10,2,3)
256

sage.coding.code_bounds.volume_hamming(n, q, r)

Returns number of elements in a Hamming ball of radius r in $$\GF{q}^n$$. Agrees with Guava’s SphereContent(n,r,GF(q)).

EXAMPLES:

sage: volume_hamming(10,2,3)
176


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