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)#

Return 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_package_guava
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_package_guava
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)#

Return the Elias upper bound.

Return the Elias upper bound for number of elements in the largest code of minimum distance \(d\) in \(\GF{q}^n\), cf. [HP2003]. If algorithm="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_package_guava
232
sage.coding.code_bounds.entropy(x, q=2)#

Compute 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()    # optional - sage.symbolic
1/10*(log(3) - 4*log(4/5) - log(1/5))/log(2)
sage: codes.bounds.entropy(1, 3)              # optional - sage.symbolic
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)#

Return the Gilbert-Varshamov lower bound.

Return 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)#

Return the Griesmer upper bound.

Return 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_package_guava
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)#

Return the Hamming upper bound.

Return 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.

\[M \leq \frac{q^n}{V(n,e)},\]

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)#

Return the Plotkin upper bound.

Return 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_package_guava
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)#

Return the Singleton upper bound.

Return 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)#

Return the number of elements in a Hamming ball.

Return 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