And similarly, five byte values start with $110\,01$, representing values in the range $2^{27}\le n \lt 2^{35}$, six byte values start with $110\,10$, representing values in the range $2^{35}\le n \lt 2^{43}$, seven byte values start with $110\,11$, representing values in the range $2^{43}\le n \lt 2^{51}$.
Similarly, eight byte values start with $1110\,000$, $1110$ being unary for
a three bit wide field, and $000$ being those three bits, representing values in the range $2^{51}\le n \lt 2^{57}$, nine byte values start with $1110\,001$, representing values in the range $2^{57}\le n \lt 2^{65}$, ten byte values start with $1110\,010$, and so on and so forth to $1110\,111$ representing a fifteen byte value in the range $2^{105}\le n \lt 2^{113}$.
And so on and so forth except that for some time to come the reader implementation is going to throw an exception if it attempts to read a value larger than $2^{64}-1$
To represent unsigned integers, the header starts with a one bit for
positive quantities and a zero bit for negative quantities, to ensure that
bytestring order on the representation agrees with the order of the values
represented. Following that bit, we proceed as for unsigned integers, except
that for negative values, the header bits are inverted, to get correct
sort order, so that more negative values sort before less negative values
and more positive values sort after less positive values.
Thus a single byte value, representing positive signed integers in the
range $0\le n \lt 2^6$ starts with a leading one bit, followed by zero bit
(unary for a zero width count field), and a single byte value representing
negative integers in the range $-2^6\le n \lt 0$ starts with a leading
zero bit, representing the sign, and a leading one bit, being inverted unary
for a zero width count field
Thus the representation for a signed integer $n$ in the range $-2^6\le n \lt 2^6$
is the one byte signed integer itself $\oplus 0\text{x}\,80$
A two byte value, representing signed integers in the range $2^6\le n \lt 2^{12}$ starts with the bits $1\,10\, 0$, 10 being unary for field one bit long,
as for unsigned integers
A two byte value, representing signed integers in the range $-2^{12}\le n \lt -2^6$,
starts with the bits with the bits $0\,01\, 1$, as for unsigned integers but inverted.
Thus the representation for a signed integer $n$ in the range $(-2^{12}\le n \lt 2^6)\,\lor\,(2^6\le n \lt 2^{12})$
is the two byte signed integer itself $\oplus 0\text{x}\,c000$
in big endian format.
With, as usual "not an integer value" exceptions being thrown if the central excluded range is violated.
And, similarly, the representation for a signed integer $n$ in the range $(-2^{20}\le n \lt 2^{12})\,\lor\,(2^{12}\le n \lt 2^{20})$ is the three byte signed integer itself
$\oplus 0\text{x}\,c4\,0000$
in big endian format.
And so on and so forth for signed integers of unlimited size.
QR codes and prefix free number encoding is useful in cases where we want data to be self describing – this bunch of bits is to be interpreted in a certain way, used in a certain action, means one thing, and not another thing. At present there is no standard for self description. QR codes are given meanings by the application, and could carry completely arbitrary data whose meaning and purpose comes from outside, from the context.
Ideally, it should make a connection, and that connection should then launch an interactive environment – the url case, where the url downloads a javascript app to address a particular database entry on a particular host.
A fixed length field is always in danger of
running out, so one needs a committee to allocate numbers.
With an arbitrary length field there is always plenty of
headroom, we can just let people use what numbers seem good
to them, and if there is a collision, well, one or both of
the colliders can move to another number.
For example, the hash of a public key structure has to contain an algorithm
identifier as to the hashing algorithm, to accommodate the possibility that
in future the existing algorithm becomes too weak, and we must introduce
new algorithms while retaining compatibility with the old. But there could
potentially be quite a lot of algorithms, though in practice initially there
will only be one, and it will be a long time before there are two.
When I say "arbitrarily large" I do not mean arbitrarily large, since this creates the possibility that someone could break something by sending a number bigger than the software can handle. There needs to be an absolute limit, such as sixty four bits, on representable numbers. But the limit should be larger than is ever likely to have a legitimate use.
In this class of solutions, numbers are embedded as variable sized groups of bits within a bitstream, in a way that makes it possible to find the boundary between one number and the next. It is used in data compression, but seldom used in compressed data transmission, because far too slow.
This class of problem is that of a
[universal code for integers](http://en.wikipedia.org/wiki/Universal_code_%28data_compression%29).
The particular coding I propose here is a variation on
Elias encoding, though I did not realize it when I
invented it.
On reflection, my proposed encoding is too clever by half,
better to use Elias δ coding, with large arbitrary
limits on the represented numbers, rather than
clever custom coding for each field. For the intended purpose of wrapping packets, of collecting UDP packets into messages, and messages into channels, limit the range of representable values to the range j: 0 \< j \< 2\^64, and pack all the fields representing the place of this UDP package in a bunch of messages in a bunch of channels into a single bitstream header that is then rounded into an integral number of bytes..
We have two bitstream headers, one of which contains always starts with the number 5 to identify the protocol. (Unknown protocols immediately ignored), and then another number to identify the encryption stream and the position in the encryption stream (no windowing). Then we decrypt the rest of the packet starting on a byte boundary. The decrypted packet then has additional bitstream headers.
For unsigned integers, we restrict the range to less than 2\^64-9. We then add 8 before encoding, and subtract 8 after encoding, so that our Elias δ encoded value always starts with two zero bits, which we always throw away. Thus the common values 0 to 7 inclusive are represented by a six bit value – I want to avoid wasting too much implied probability on the relatively low probability value of zero.
The restriction on the range is apt to produce unexpected errors, so I suppose we special case the additional 8 values, so that we can represent every signed integer.
For signed integers, we convert to an unsigned integer\
And then represent as a positive integer. The decoding algorithm has to know whether to call the routine for signed or unsigned. By using unsigned maths where values must always be positive, we save a bit. Which is a lot of farting around to save on one bit.
We would like a way to represent an arbitrarily large
number, a Huffman style representation of the
numbers. This is not strictly Huffman encoding,
since we want to be able to efficiently encode and decode
large numbers, without using a table, and we do not have
precise knowledge of what the probabilities of numbers are
likely to be, other than that small numbers are
substantially more probable than large numbers. In
the example above, we would like to be able to represent
numbers up to O(2^32^), but efficiently represent
the numbers one, and two, and reasonably efficiently
represent the numbers three and four. So to be
strictly correct, “prefix free number encoding”. As we
shall see at the end, prefix free number encoding always
corresponds to Huffman encoding for some reasonable weights
– but we are not worrying too much about weights, so are
not Huffman encoding.
###Converting to and from the representation
Assume X is a prefix free sequence of bit strings – that is to say, if we
are expecting a member of this sequence, we can tell where the member
ends.
Let \[m…n\] represent a sequence of integers m to n-1.
Then the function X→\[m…n\] is the function that converts a bit string of X
to the corresponding integer of \[m…n\], and similarly for \[m…n\]→X.
Thus X→\[m…n\] and \[m…n}→X provide us with a prefix free representation of
numbers greater than or equal to m, and less than n.
Assume the sequence X has n elements, and we can generate and recognize
each element.
Let ℓ(X,k) be a new sequence, constructed by taking the first element of
X, and appending to it the 2^k^ bit patterns of length i, the
next element of X and appending to it the 2^k+1^ bit patterns of
length k+1, and so on and so forth.
ℓ is a function that gives us this new sequence from an existing sequence
and an integer.
The new sequence ℓ(X,k) will be a sequence of prefix free bit patterns
that has 2^n+k+1^-2^k^ elements.
We can proceed iteratively, and define a sequence ℓ(ℓ(X,j),k), which class
of sequences is useful and efficient for numbers that are typically quite
small, but could often be very large.We will more precisely
prescribe what sequences are useful and efficient for what purposes when
we relate our encoding to Huffman coding.
To generate the m+1[th]{.small} element of ℓ(X,k), where X is a
sequence that has n elements:
Let j = m + 2^k^
Let p = floor(log~2~(j)) that is to say, p is the position of
the high order bit of j, zero if j is one, one if j is two
or three, two if j is four, five, six, or seven, and so on and so forth.
We encode p into its representation using the encoding \[k…n+k\]→X, and
append to that the low order p bits of j.
To do the reverse operation, decode from the prefix free representation to
the zero based sequence position, to perform the function ℓ(X,k)→\[0…2^n+k+1^-2^k^\],
we extract p from the bit stream using the decoding of X→\[j…n+j\], then
extract the next p bits of the bit stream, construct k from 2^p^-2^j^
plus the number represented by those bits.
Now all we need is an efficient sequence X for small numbers.
Let ℒ(n) be a such a sequence with n values.\
The first bit pattern of ℒ(n) is 0\
The next bit pattern of ℒ(n) is 10\
The next bit pattern of ℒ(n) is 110\
The next bit pattern of ℒ(n) is 1110\
…\
The next to last bit pattern of ℒ(n) is 11…110, containing n-2 one bits
and one zero bit.\
The last bit pattern of ℒ(n) breaks the sequence, for it is 11…11,
containing n-1 one bits and no zero bit.
The reason why we break the sequence, not permitting the
representation of unboundedly large numbers, is that
computers cannot handle unboundedly large numbers – one
must always specify a bound, or else some attacker will
cause our code to crash, producing results that we did not
anticipate, that the attacker may well be able to make use
of.
Perhaps a better solution is to waste a bit, thereby
allowing future expansion. We use a representation
that can represent arbitrarily large numbers, but clients
and servers can put some arbitrary maximum on the size of
the number. If that maximum proves too low, future clients
can just expand it without breaking backward compatibility.
This is similar to the fact that different file systems
have different arbitrary maxima for the nesting of
directories, the length of paths, and the length of
directory names. Provided the maxima are generous
it does not matter that they are not the same.
Thus the numbers 1 to 2 are represented by \[1…3\] →
ℒ(2), 1 being the pattern “0”, and 2 being the
pattern “1”
The numbers 0 to 5 are represented by \[0…6\] → ℒ(6), being the patterns\
“0”, “10”, “110”, “1110”, “11110”, “11111”
Thus \[0…6\] → ℒ(6)(3) is a bit pattern that represents the number
3, and it is “1110”
This representation is only useful if we expect our numbers
to be quite small.
\[0…6\] → ℓ(ℒ(2),1) is the sequence “00”, “01”,
“100”, “101”, “110”, “111” representing the
numbers zero to five, representing the numbers 0 to
representing the numbers one to fourteen, representing the
numbers 1 to less than 1 + 2^3+1^ – 2^1^
We notice that ℓ(ℒ(n),k) has 2^n+k^ – 2^k^
patterns, and the shortest patterns are length 1+k, and the
largest patterns of length 2n+k-2
This representation in general requires twice as many bits
as to represent large numbers as the usual, non self
terminating representation does (assuming k to be small)
We can iterate this process again, to get the bit string sequence:\
ℓ(ℓ(ℒ(n),j),k)\
which sequence has 2\^(2^n+j^-2^j^+k)-2^k^
elements.
This representation is asymptotically efficient for very
large numbers, making further iterations pointless.
ℓ(ℒ(5),1) has 62 elements, starting with a two bit pattern, and ending
with a nine bit pattern.Thus ℓ(ℓ(ℒ(5),1),2) has
2^64^-4 elements, starting with a four bit pattern, and finishing
with a 72 bit pattern.
### prefix free encoding as Huffman coding
Now let us consider a Huffman representation of the
numbers when we assign the number `n` the
weight `1/(n*(n+1)) = 1/n – 1/(n+1)`
In this case the weight of the numbers in the range `n ... m` is `1/n – 1/(m+1)`
So our bit patterns are:\
0 (representing 1)\
100 101 representing 2 to 3\
11000 11001 11010 11011 representing 4 to 7\
1110000 1110001 1110010 1110011 1110100 1110101
1110110 1110111 representing 8 to 15
We see that the Huffman coding of the numbers that are
weighted as having probability `1/(n*(n+1))`
Is our friend \[1…\] → ℓ(ℒ(n),0), where n is very large.
Thus this is good in a situation where we are quite unlikely to encounter
a big number. However a very common situation, perhaps the most
common situation, is that we are quite likely to encounter numbers smaller
than a given small amount, but also quite likely to encounter numbers
larger than a given huge amount – that the probability of encountering a
number in the range 0…5 is somewhat comparable to the probability of
encountering a number in the range 5000…50000000.
We want an encoding that corresponds to a Huffman encoding where numbers are logarithmically distributed up to some enormous limit, corresponding to an encoding where for all n, n bit numbers are represented with an only slightly larger number of bits, n+O(log(n)) bits.
In such case, we should we should represent such values by members of a
