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6
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1answer
97 views

Lower bound on prefix code lengths

For a prefix code $C:\{0,1\}^*\to\{0,1\}^*$, define $f(n)$ as the length of the longest encoding of a number with up to $n$ bits: $$ f(n)=\max_{|k|\le n}\left|C(k)\right|. $$ (Note that by taking ...
1
vote
1answer
150 views

Does Huffman coding always produce shorter codes than the Shannon code?

Let $X\in\{1,2,\ldots,m\}$ be a discrete random variable with $X\sim p$. Let $C$ be a code for $X$ with $l_i$ being the length $i$-th codeword and let $L(C)$ be the expected length of the code. The ...
7
votes
0answers
68 views

What is the optimal binary encoding of the elements of a monoid?

The Question Let $M$ be a finite monoid. Let $S$ be a generating set of $M$. Say we have a binary encoding of $S$ represented by $\phi:S \rightarrow A^*$ where $A = \{0, 1\}$. This encoding should ...
5
votes
1answer
326 views

NP-hardness of minimum distance over a code

There happens to be this NP-complete question, Minimum-Distance-Over-$\mathbb{F}_{2^m}$ Given $w \in \mathbb{Z}^+$ and a $r \times n$ matrix $H$ over $\mathbb{F}_{2^m}$, is there a $x \in ...
1
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0answers
90 views

Existence of incremental algorithm for (canonical) prefix-free codes

Short version Given the $N$ code lengths of an optimal prefix-free code $L[1..N]$ for $N$ unknown weights $W[1..N]$ of known sum $S$, and $N'$ new weights $W'[1..N']$ all larger than the weights ...
8
votes
2answers
274 views

How do I construct an optimal affix code?

An affix code is a code that is simultaneously a prefix and suffix code. That is, no codeword is neither the prefix nor the suffix of any other codeword. Affix codes can be instantaneously decoded in ...
1
vote
1answer
814 views

Definition of a prefix-free Turing machine

A prefix-free function is one whose domain is prefix-free. Similarly, a prefix-free (Turing) machine is one whose domain is prefix-free. It is usual to consider such a machine as being ...
5
votes
0answers
153 views

What is the best upper bound known for the complexity of computing an optimal prefix free code in the RAM model?

In the algebraic decision tree, the result is clear: Elmasry and Belal (Verification Of Minimum Redundancy Prefix Codes)'s lower bound shows that the worst case complexity of computing an optimal ...