r"""
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.
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
.. math::
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
.. math::
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
.. math::
\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 minimum_distance_lower_bound(n,k,F) and
minimum_distance_upper_bound(n,k,F) (both of which which connect
to the internet using Steven Sivek's linear_code_bound(q,n,k))
(2) codesize_upper_bound(n,d,q), dimension_upper_bound(n,d,q).
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.
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.
"""
from sage.interfaces.all import gap
from sage.rings.all import QQ, RR, ZZ, RDF
from sage.rings.arith import factorial
from sage.functions.all import log, sqrt
from sage.misc.decorators import rename_keyword
@rename_keyword(deprecated='Sage version 4.6', method="algorithm")
def codesize_upper_bound(n,d,q,algorithm=None):
r"""
This computes the minimum value of the upper bound using the
algorithms 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 UpperBound( n, d, q ).
EXAMPLES::
sage: codesize_upper_bound(10,3,2)
93
sage: codesize_upper_bound(10,3,2,algorithm="gap") # requires optional GAP package Guava
85
"""
if algorithm=="gap":
return int(gap.eval("UpperBound(%s,%s,%s)"%( n, d, q )))
else:
eub = elias_upper_bound(n,q,d)
gub = griesmer_upper_bound(n,q,d)
hub = hamming_upper_bound(n,q,d)
pub = plotkin_upper_bound(n,q,d)
sub = singleton_upper_bound(n,q,d)
return min([eub,gub,hub,pub,sub])
def dimension_upper_bound(n,d,q):
r"""
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.
EXAMPLES::
sage: dimension_upper_bound(10,3,2)
6
"""
q = ZZ(q)
return int(log(codesize_upper_bound(n,d,q),q))
def volume_hamming(n,q,r):
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
"""
ans=sum([factorial(n)/(factorial(i)*factorial(n-i))*(q-1)**i for i in range(r+1)])
return ans
def gilbert_lower_bound(n,q,d):
r"""
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
"""
ans=q**n/volume_hamming(n,q,d-1)
return ans
@rename_keyword(deprecated='Sage version 4.6', method="algorithm")
def plotkin_upper_bound(n,q,d, algorithm=None):
r"""
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") # requires optional GAP package Guava
192
"""
if algorithm=="gap":
ans=gap.eval("UpperBoundPlotkin(%s,%s,%s)"%(n,d,q))
return QQ(ans)
else:
t = 1 - 1/q
if (q==2) and (n == 2*d) and (d%2 == 0):
return 4*d
elif (q==2) and (n == 2*d + 1) and (d%2 == 1):
return 4*d + 4
elif d > t*n:
return int(d/( d - t*n))
elif d < t*n + 1:
fact = (d-1) / t
if RR(fact)==RR(int(fact)):
fact = int(fact) + 1
return int(d/( d - t * fact)) * q**(n - fact)
@rename_keyword(deprecated='Sage version 4.6', method="algorithm")
def griesmer_upper_bound(n,q,d,algorithm=None):
r"""
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") # requires optional GAP package Guava
128
"""
if algorithm=="gap":
ans=gap.eval("UpperBoundGriesmer(%s,%s,%s)"%(n,d,q))
return QQ(ans)
else:
den = 1
s = 0
k = 0
add = 0
while s <= n:
if not(add == 1):
if d%den==0:
add = int(d/den)
else:
add = int(d/den)+1
s = s + add
den = den * q
k = k + 1
return q**(k-1)
@rename_keyword(deprecated='Sage version 4.6', method="algorithm")
def elias_upper_bound(n,q,d,algorithm=None):
r"""
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") # requires optional GAP package Guava
232
"""
r = 1-1/q
if algorithm=="gap":
ans=gap.eval("UpperBoundElias(%s,%s,%s)"%(n,d,q))
return QQ(ans)
else:
def ff(n,d,w,q):
return r*n*d*q**n/((w**2-2*r*n*w+r*n*d)*volume_hamming(n,q,w));
def get_list(n,d,q):
I = []
for i in range(1,int(r*n)+1):
if i**2-2*r*n*i+r*n*d>0:
I.append(i)
return I
I = get_list(n,d,q)
bnd = min([ff(n,d,w,q) for w in I])
return int(bnd)
def hamming_upper_bound(n,q,d):
r"""
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.
.. math::
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
"""
return int((q**n)/(volume_hamming(n, q, int((d-1)/2))))
def singleton_upper_bound(n,q,d):
r"""
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
.. math::
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
"""
return q**(n - d + 1)
def 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))
0.367049926083
"""
q = ZZ(q)
ans=log(gilbert_lower_bound(n,q,int(n*delta)),q)/n
return ans
def entropy(x,q):
"""
Computes the entropy at `x` on the `q`-ary symmetric channel.
INPUT:
- ``x`` - real number in the interval `[0, 1]`.
- ``q`` - 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(1/5)/log(4) - 4/5*log(4/5)/log(4) + 1/5*log(3)/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
"""
if x < 0 or x > 1:
raise ValueError("The entropy function is defined only for x in the"
" interval [0, 1]")
q = ZZ(q)
if q < 2:
raise ValueError("The value q must be an integer greater than 1")
if x == 0:
return 0
if x == 1:
return log(q-1,q)
H = x*log(q-1,q)-x*log(x,q)-(1-x)*log(1-x,q)
return H
def gv_bound_asymp(delta,q):
"""
Computes the asymptotic GV bound for the information rate, R.
EXAMPLES::
sage: RDF(gv_bound_asymp(1/4,2))
0.188721875541
sage: f = lambda x: gv_bound_asymp(x,2)
sage: plot(f,0,1)
"""
return (1-entropy(delta,q))
def hamming_bound_asymp(delta,q):
"""
Computes the asymptotic Hamming bound for the information rate.
EXAMPLES::
sage: RDF(hamming_bound_asymp(1/4,2))
0.4564355568
sage: f = lambda x: hamming_bound_asymp(x,2)
sage: plot(f,0,1)
"""
return (1-entropy(delta/2,q))
def 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)
"""
return (1-delta)
def 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
"""
r = 1-1/q
return (1-delta/r)
def 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...
"""
r = 1-1/q
return RDF((1-entropy(r-sqrt(r*(r-delta)), q)))
def 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)
0.354578902665
"""
return RDF(entropy((q-1-delta*(q-2)-2*sqrt((q-1)*delta*(1-delta)))/q,q))
