A method to find the volume of a sphere in the Lee metric, and its applications

Sagnik Bhattacharya, Adrish Banerjee (IIT Kanpur)

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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?

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0

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2 2

33

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What can we do with the answer?

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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.

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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

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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

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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

?

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Involving a probability distributionWe start by dividing both sides of by

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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

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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 .

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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.

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Sanity check for the Hamming metric

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Convex OptimisationIn general, we need to use convex optimisation to find P*

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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.

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Convex optimisation - functional formThe dual program is given by

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Convex optimisation - functional formThe dual program is given by

Since any will give an upper bound by strong duality,

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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.

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What is the value of ?

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Immediate result - asymptotic sizes of Lee balls

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Also - bounds on codes in Lee metric

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Key takeaways● Solving an algebraic problem using analytic techniques

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Key takeaways● Solving an algebraic problem using analytic techniques

● Method generalises to all discrete metrics with the following

property

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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

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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.

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Thank you!

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