Probabilistic Methods in Coding Theory: Asymmetric Covering Codes. Joshua N. Cooper UCSD Dept. of Mathematics Robert B. Ellis Texas A&M Dept. of Mathematics Andrew B. Kahng UCSD Dept. of Computer Science and Engineering. Covering Codes.
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Joshua N. Cooper
UCSD Dept. of Mathematics
Robert B. Ellis
Texas A&M Dept. of Mathematics
Andrew B. Kahng
UCSD Dept. of Computer Science and Engineering
Definition: A code is a subset of the set of binary nstrings. The (Hamming) distance d(S,T) between two strings S and T is the number of coordinates at which they differ.
Definition: A covering code of radius R is a code C such that, for any binary nstring S, d(S,c) <= R for some element c in C. We use the convention that R will always be as small as possible.
Definition: An asymmetric covering code of radius R is a code C such that, for any binary nstring S, there is some element c of C which can be turned into S by flipping at most R 0’s to 1’s.
Asymmetric LossWhat if only certain types of loss are allowed? For example, we can prohibit changes from 1’s to 0’s  invalidating the 1010 > 1110 change from before.
Define the (asymmetric) ball of radius R about the vector (string) x to be the set of strings reachable from the vector x by changing at most R ones to zeroes.
A set of vectors is a asymmetric covering code of radius R iff the
balls centered at the codewords cover the ncube (and R is the
smallest integer so that this is true).
2. nR (the “coradius”)is constant.
3. Everything else.
QuestionsHow large is the smallest possible asymmetric covering code of radius R on n bits? Call this number K+(n,R).
Interesting cases:



V3 (3;1) = 4
V3 (2;1) = 3
V3 (1;1) = 2
V3 (0;1) = 1
SphereCovering Lower Bound: Ball SizesSize of symmetric ball: independent of ball center
Size of asymmetric ball: determined by weight of ball center
Definition. Vn(R): size of symmetric ball of radius R in Qn
Vn (k;R): size of asymmetric ball of radius R about a point of weight k


Vn(R) =
Vn (k;R) =
Theorem (Asymmetric SphereCovering Bound).
Proof. There are vertices of weight k. Fix such a vertex.
Largest ball containing such a vertex is centered at level k+R
The vertex contributes at least 1/ to the size of K+(n,R).
n
k
( )
K+(n,R)

Vn (k+R;R)
SphereCovering Lower Bound (Constant Radius)Theorem (Symmetric SphereCovering Bound).
K(n,R)
How close can we come to the spherecovering lower bound?
Theorem: The size of a minimal (symmetric) covering code of
constant radius R is θ(2n/nR).
The construction is linear, i.e., produces a code which is a linear
subspace of GF(2)n.
This is a problem, because the algebraic structure of Hamming
space shows no respect at all for the asymmetric covering relation.
In fact, minimal linear asymmetric covering codes turn out to be
HUGE, like θ(2n).
Let’s try choosing a set of codewords randomly. Let the
probability of picking a point x be proportional to log n divided by
the number of points which x covers.
Then, throw in anything we missed.
Result: A code of size θ(2n log n/nR). Close, but no cigar.
Proposition: is an asymmetric covering code of radius
R+S if C1 and C2 are asymmetric covering codes of radius R
and S, respectively, so, in particular,
K+(n+m,R+S) <= K+(n,R) K+(m,S).
A ToolFor asymmetric covering codes C1 and C2, define the direct sum to be all concatenations of strings from C1 and C2:
Write the cube as a product of two cubes of about the same
dimension.
1. Perform the previous random process on the first factor.
2. For each red point, put a full copy of the second factor in.
3. For each blue point, put an asymmetric covering code of
the second factor in. (I.e., induction!)
Theorem. radiusK+(n,R) = θ(2n/nR).
Next question: What’s the constant?
Define μ*(R) = lim inf (K(n,R) * volume of Rball / 2n)
= min symmetric “redundancy” factor.
Define μ0*(R) = lim inf (K+(n,R) * average vol. of Rball / 2n)
= min asymmetric “redundancy” factor.
Did it help???? radius
???
Open QuestionsThis construction gives an exponential upper bound for μ0*(R) and
μ*(R). Van Vu has given an upper bound of cR log R for both.
Conjecture. μ*(R) = 1 for all R.
But what is μ0*(R)?
Question: Are there efficiently computable minimal asymmetric
covering codes?
Question: What is the order of K+(n,R) where R is not constant?
Question: What about the nonbinary case?
Best Known Bounds radius
A single value appears if the bound is tight;
otherwise a range of values appears.