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A method to find the volume of a sphere in the Lee metric, and its applications
Sagnik Bhattacharya, Adrish Banerjee (IIT Kanpur)
1
ISIT 2019July 9, 2019
Question
How do we find bounds on the size of codes for discrete metrics
other than the Hamming metric? What about the Lee metric?
2
0
11
2 2
33
4
What can we do with the answer?
3
Notation - The volume of a Lee ball of radius and in a code with
blocklength is given by
What can we do with the answer?
41. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)
Notation - The volume of a Lee ball of radius and in a code with
blocklength is given by
● Chiang and Wolf1 gave the following bounds for the Lee
metric
What can we do with the answer?
51. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)
Notation - The volume of a Lee ball of radius and in a code with
blocklength is given by
● Chiang and Wolf1 gave the following bounds for the Lee
metric
What can we do with the answer?
61. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)
Hamming Bound
Notation - The volume of a Lee ball of radius and in a code with
blocklength is given by
● Chiang and Wolf1 gave the following bounds for the Lee
metric
What can we do with the answer?
71. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)
Hamming Bound Gilbert-Varshamov Bound
Notation - The volume of a Lee ball of radius and in a code with
blocklength is given by
● Chiang and Wolf1 gave the following bounds for the Lee
metric
● So if we know we will also know bounds on
What can we do with the answer?
Hamming Bound Gilbert-Varshamov Bound
81. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)
Notation - The volume of a Lee ball of radius and in a code with
blocklength is given by
● Chiang and Wolf1 gave the following bounds for the Lee
metric
● So if we know we will also know bounds on
● Berlekamp2 also gave the EB bound for the Lee metric
What can we do with the answer?
Hamming Bound Gilbert-Varshamov Bound
91. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)2. R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
So the question is to figure out
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So the question is to figure out ● Chiang and Wolf3 gave the following expression.
113. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)
So the question is to figure out ● Chiang and Wolf3 gave the following expression
which is mathematically intractable.
123. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)
So the question is to figure out ● Chiang and Wolf3 gave the following expression
which is mathematically intractable.
● Roth4 gave the following expression for radius
133. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)4. R. Roth, Introduction to Coding Theory (2006)
So the question is to figure out ● Chiang and Wolf3 gave the following expression
which is mathematically intractable.
● Roth4 gave the following expression for radius
But the parameter regime is too small.
143. J. C.-Y. Chiang and J. K. Wolf, “On channels and codes for the Lee metric” (1971)4. R. Roth, Introduction to Coding Theory (2006)
Our contribution(s)● We use the generating function for a metric and Sanov’s
theorem to find the volume of a sphere of given radius.
15
Our contribution(s)● We use the generating function for a metric and Sanov’s
theorem to find the volume of a sphere of given radius.
● It reduces to the known results for the Hamming metric
16
Our contribution(s)● We use the generating function for a metric and Sanov’s
theorem to find the volume of a sphere of given radius.
● It reduces to the known results for the Hamming metric
● It allows us to find bounds on the rate for the Lee metric
17
Our Methods
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19Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
20Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● The generating function for the
21Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● Because the distance function is additive over the
coordinates…
22Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● Because the distance function is additive over the
coordinates…
● The generating function is multiplicative
23Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● Because the distance function is additive over the
coordinates…
● The generating function is multiplicative and
24Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● Because the distance function is additive over the
coordinates…
● The generating function is multiplicative and
● gives the weights for a single symbol only.
25Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● Because the distance function is additive over the
coordinates…
● The generating function is multiplicative and
● gives the weights for a single symbol only.
q-ary Hamming Metric
26Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● Because the distance function is additive over the
coordinates…
● The generating function is multiplicative and
● gives the weights for a single symbol only.
q-ary Hamming MetricLee metric for odd q
27Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
The Generating Function● Because the distance function is additive over the
coordinates…
● The generating function is multiplicative and
● gives the weights for a single symbol only.
q-ary Hamming MetricLee metric for odd q
Lee metric for even q28Ref - R. Berlekamp, Algebraic Coding Theory - Revised Edition (2015)
New Question
How do we find
?
29
Involving a probability distributionWe start by dividing both sides of by
30
Involving a probability distributionWe start by dividing both sides of by to get
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Involving a probability distributionWe start by dividing both sides of by to get
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Involving a probability distributionSay we start from
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Involving a probability distributionSay we start from
Dividing by q, we get
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Involving a probability distributionSay we start from
Dividing by q, we get
Which defines a discrete random variable that takes value 0
w.p. 1/q, 1 w.p. 2/q and so on.
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Involving a probability distribution
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Involving a probability distribution
The give the probability that the sum of i.i.d. samples
drawn according to add up to
37
Sanov’s Theorem
38Sanov, I. N. (1957) "On the probability of large deviations of random variables"Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory (2 ed.)
Using Sanov’s theorem - finding EThe natural choice while calculating the coefficient is the set
of all distributions with mean less than or equal to .
39
Sanity Check - Hamming metricThe Hamming metric does not require convex optimisation. The
random variable in the Hamming case is Bernoulli, and the KL
divergence minimising distribution is not hard to find. The result
is familiar.
40
Sanity check for the Hamming metric
41
Convex OptimisationIn general, we need to use convex optimisation to find P*
42
Convex OptimisationIn general, we need to use convex optimisation to find P*
Strong duality holds for the problem, so any solution to the dual
implies an upper bound for the primal.
43
Convex optimisation - functional formThe dual program is given by
44
Convex optimisation - functional formThe dual program is given by
Since any will give an upper bound by strong duality,
45
Convex optimisation - functional formThe dual program is given by
Since any will give an upper bound by strong duality, we can
choose the function to be
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Functional form - intuition
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Functional form - intuition
● should be zero
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Functional form - intuition
● should be zero
● should be monotonically increasing
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Functional form - intuition
● should be zero
● should be monotonically increasing
Open - more analytical justification of why this is the form.
50
What is the value of ?
51
Immediate result - asymptotic sizes of Lee balls
52
Also - bounds on codes in Lee metric
53
Key takeaways● Solving an algebraic problem using analytic techniques
54
Key takeaways● Solving an algebraic problem using analytic techniques
● Method generalises to all discrete metrics with the following
property
55
Key takeaways● Solving an algebraic problem using analytic techniques
● Method generalises to all discrete metrics with the following
property - the set of distances of all symbols to one fixed
symbol remains the same when the fixed symbol is replaced
by some other symbol
56
Key takeaways● Solving an algebraic problem using analytic techniques
● Method generalises to all discrete metrics with the following
property - the set of distances of all symbols to one fixed
symbol remains the same when the fixed symbol is replaced
by some other symbol
● Should generalise to other discrete metrics too, but the
expressions would be more complicated.
57
Thank you!
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