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 set of size \(q\). A subset \(C\) of \(V=F^n\) is called a code of length \(n\). Often one considers the case where \(F\) is a finite field, denoted by \(\GF{q}\). Then \(V\) is an \(F\)-vector space. 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 elements of \(V=F^n\) 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 an element (in the Hamming metric) is \(d({\bf v},{\bf 0})\), where \(0\) is a distinguished element of \(F\); in particular it is \(0\) of the field if \(F\) is a field. 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 \(A_q(n,d)\) 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. For more details see [HP2003] and [Lin1999].

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.

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

Returns an upper bound on the number of codewords in a (possibly non-linear) code.

This function computes the minimum value of the upper bounds 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: codes.bounds.codesize_upper_bound(10,3,2)
93
sage: codes.bounds.codesize_upper_bound(24,8,2,algorithm="LP")
4096
sage: codes.bounds.codesize_upper_bound(10,3,2,algorithm="gap")  # optional - gap_packages (Guava package)
85
sage: codes.bounds.codesize_upper_bound(11,3,4,algorithm=None)
123361
sage: codes.bounds.codesize_upper_bound(11,3,4,algorithm="gap")  # optional - gap_packages (Guava package)
123361
sage: codes.bounds.codesize_upper_bound(11,3,4,algorithm="LP")
109226

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

Return an upper bound for the dimension of a linear code.

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

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

The asymptotic Elias bound for the information rate.

This only makes sense when \(0 < \delta < 1-1/q\).

EXAMPLES:

sage: codes.bounds.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.

Returns the Elias upper bound for number of elements in the largest code of minimum distance \(d\) in \(\GF{q}^n\), cf. [HP2003]. If the method is “gap”, it wraps GAP’s UpperBoundElias.

EXAMPLES:

sage: codes.bounds.elias_upper_bound(10,2,3)
232
sage: codes.bounds.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: codes.bounds.entropy(0, 2)
0
sage: codes.bounds.entropy(1/5,4).factor()
1/10*(log(3) - 4*log(4/5) - log(1/5))/log(2)
sage: codes.bounds.entropy(1, 3)
log(2)/log(3)

Check that values not within the limits are properly handled:

sage: codes.bounds.entropy(1.1, 2)
Traceback (most recent call last):
...
ValueError: The entropy function is defined only for x in the interval [0, 1]
sage: codes.bounds.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)
0.012986862055...
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 the Gilbert-Varshamov lower bound.

Returns the Gilbert-Varshamov lower bound for number of elements in a largest code of minimum distance d in \(\GF{q}^n\). See Wikipedia article Gilbert-Varshamov_bound

EXAMPLES:

sage: codes.bounds.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.

Returns the Griesmer upper bound for the number of elements in a largest linear code of minimum distance \(d\) in \(\GF{q}^n\), cf. [HP2003]. If the method is “gap”, it wraps GAP’s UpperBoundGriesmer.

The bound states:

\[`n\geq \sum_{i=0}^{k-1} \lceil d/q^i \rceil.`\]

EXAMPLES:

The bound is reached for the ternary Golay codes:

sage: codes.bounds.griesmer_upper_bound(12,3,6)
729
sage: codes.bounds.griesmer_upper_bound(11,3,5)
729

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

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

The asymptotic Gilbert-Varshamov bound for the information rate, R.

EXAMPLES:

sage: RDF(codes.bounds.gv_bound_asymp(1/4,2))
0.18872187554086...
sage: f = lambda x: codes.bounds.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)

The Gilbert-Varshamov lower bound for information rate.

The Gilbert-Varshamov lower bound for information rate of a \(q\)-ary code of length \(n\) and minimum distance \(n\delta\).

EXAMPLES:

sage: RDF(codes.bounds.gv_info_rate(100,1/4,3))  # abs tol 1e-15
0.36704992608261894

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

The asymptotic Hamming bound for the information rate.

EXAMPLES:

sage: RDF(codes.bounds.hamming_bound_asymp(1/4,2))
0.456435556800...
sage: f = lambda x: codes.bounds.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.

Returns the Hamming upper bound for number of elements in the largest code of length n and minimum distance d over alphabet of size q.

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 an alphabet of size \(q\). The Hamming bound is obtained by dividing the contents of the entire Hamming space \(q^n\) 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. See e.g. [HP2003].

EXAMPLES:

sage: codes.bounds.hamming_upper_bound(10,2,3)
93

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

The first asymptotic McEliese-Rumsey-Rodemich-Welsh bound.

This only makes sense when \(0 < \delta < 1-1/q\).

EXAMPLES:

sage: codes.bounds.mrrw1_bound_asymp(1/4,2)   # abs tol 4e-16
0.3545789026652697

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

The asymptotic Plotkin bound for the information rate.

This only makes sense when \(0 < \delta < 1-1/q\).

EXAMPLES:

sage: codes.bounds.plotkin_bound_asymp(1/4,2)
1/2

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

Returns the Plotkin upper bound.

Returns the Plotkin upper bound for the number of elements in a largest code of minimum distance \(d\) in \(\GF{q}^n\). More precisely this is a generalization of Plotkin’s result for \(q=2\) to bigger \(q\) due to Berlekamp.

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

EXAMPLES:

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

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

The asymptotic Singleton bound for the information rate.

EXAMPLES:

sage: codes.bounds.singleton_bound_asymp(1/4,2)
3/4
sage: f = lambda x: codes.bounds.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.

Returns the Singleton upper bound for number of elements in a largest code of minimum distance d in \(\GF{q}^n\).

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: codes.bounds.singleton_upper_bound(10,2,3)
256

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

Returns the number of elements in a Hamming ball.

Returns the number of elements in a Hamming ball of radius \(r\) in \(\GF{q}^n\).

EXAMPLES:

sage: codes.bounds.volume_hamming(10,2,3)
176