动态单一位编码：二进制数据编码的新境界
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"动态单一位编码（2014年12月19日）- 计算机科学"
本文档是Ernst D. Berg撰写的一篇关于动态单一位编码（Dynamic Unary Encoding，简称DUE）的介绍。动态单一位编码是对传统单一位编码的扩展，它利用了二进制字符串和奇偶校验信息，将2^n个非负整数划分为n个互不相交的n位元素循环集合。这些循环在虚拟数据集、二进制变换和数学对象中都有应用。
1. 动态单一位编码的概念
动态单一位编码是一种数据编码方法，最初由作者在2010年尝试通过迭代函数实验性二进制编码方案解决数据压缩挑战时发现。与传统的单一位编码不同，动态单一位编码允许每个n位的二进制字符串既是编码方式，也能被该编码方式编码。
2. 编码原理
动态单一位编码结合了两种形式的单一位编码，并添加了一个奇偶校验位。这使得2^n个非负整数被划分为n个独立的循环，每个循环包含n位的元素。这些循环集合具有重要的特性，即它们是互斥的，这意味着每个整数只属于其中一个循环。
3. 循环结构与频谱分析
文档中对这些循环结构及其频谱进行了特征化分析。这包括对循环本身的数学属性的理解，以及对不同循环出现频率的统计特性。
4. 编解码算法
文章提供了编码和解码算法的例子，这些算法是基于动态单一位编码的原理实现的。这些算法的目的是有效地将数字转化为动态单一位编码形式，以及将编码后的形式还原回原始数值。
5. 应用示例
除了基础的编解码算法，文档还展示了其他利用动态单一位编码原理构建的构造实例。动态单一位编码的循环结构作为一个数学对象的概念得到了充分展示，说明了其在更广泛领域的潜在应用价值。
6. 结论
动态单一位编码提供了一种新的数据表示和处理方式，它不仅提高了数据编码的效率，还在数据组织和计算上展现了独特的数学美。通过深入理解动态单一位编码的理论和实践，我们可以探索更多潜在的应用场景，特别是在数据压缩、二进制转换和相关数学领域。
2.2 Decoding
First, decoding determines the terminus parity of the unary code used to encode from the nth (b
n−1
) bit of
the string. Then the string is parsed into unary codes by terminus boundaries. Same parity segments, of
the length the unary code communicates, are then written in correct parity order using the Parity Reference
information. An example decode algorithm is presented in figure 2.
• Let -σ represent decoding.
• Let our source string be “1110110001010111110110100101101001011000”
• Let our unary code terminus be 1 and our Parity Reference = b
0
. Terminus information is read at b
n−1
.
Parse the string into unary codes 1 1 10 1 1000 10 10 1 1 1 1 10 1 10 100 10 1 10 100 10 1 1000
Same parity bit segments 0 1 00 1 0000 11 00 1 0 1 0 11 0 11 000 11 0 11 000 11 0 1111
-σ(1110110001010111110110100101101001011000)
1
= (0100100001100101011011000110110001101111)
0
3 Cycles and Cycle Spectrum
Every n-bit binary string is an encoding of dynamic unary and every n-bit binary string is encodable by
dynamic unary. Through experimentation and observation iteration of encoding or decoding resulted in
a cycle of k elements of length n bits. By utilizing both forms of unary code and the parity structure
information of the source string dynamic unary partitions 2
n
non-negative integers into disjoint cycles[3][4]
of n-bit elements creating a dynamic data type. The spectrum of these cycles is an infinite progression of
sets[10] of cycles that have in common relationships to powers of two.
2
The quality of the cycles being
disjoint was first determined by observations of the cycles of the shorter length strings which are easy
to generate by hand. The disjoint quality of the cycles was also observed with a C-Language program
generating the cycles of thirty two bit elements, as a data set, of all 2
32
non-negative integers. Encoding
is further defined as generating the next element in cycle and decoding is further defined as generating the
previous element in cycle.
S
0
= {0000} n = 4 k = 8 PRef = b
0
S
i+1
= σ(S
0
)
i
or -σ(S
0
)
i
for 0 ≥ i < k S
k
= S
0
σ(0000) = ( 0111 1100 0101 1111 1000 0011 1010 0000 ) = ( 7 12 5 15 8 3 10 0 )
-σ(0000) = ( 1010 0011 1000 1111 0101 1100 0111 0000 ) = ( 10 3 8 15 5 12 7 0 )
S
0
7→ S
1
7→ S
2
7→ S
3
7→ S
4
7→ S
5
7→ S
6
7→ S
7
= S
0
Example cycle in both the encode direction and the decode direction of a 4-bit string.
The number of cycles for a n-bit binary string and a specific Parity Reference bit position is determined by
x equals the number of elements in a cycle and 2
n
the number of non-negative integers such that cycles =
2
n
/x. That there are n choices for a Parity Reference means we also have n sets of cycles for each n-bit
string.
• From observations of the cycle data, cycle spectrum can be separated into three categories.
• String length (2
n
)
b
0
has (2
1+n
) cycles of (2
n
) elements for n = 0 .
3
• Binary string lengths that are a power of two, greater than one, and Parity Reference of b
0
have been observed
as having an element count of (2
1+n
) and cycle count of (2
2
n
/2
1+n
) for n ∈ {1, 2, 3, . . . }.
• All other lengths and Parity Reference combinations may be defined as (2
1+blog
2
nc
) elements
4
and (2
1+n
/2
1+blog
2
nc
)
cycles for n ∈ {1, 2, 3, . . .}.
2
A complete listing of the cycles for string lengths one through eight is presented in table one.
3
In researching the Mathematics of Cycles it was observed that a cycling of a single element is not generally considered a
Cycle;[3] however, what applies to a string of one bit, in DUE, applies to all strings and is therefore included as such.
4
(2
1+blog
2
nc
)
n∈{1,2,3,...}
= {2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, ...}
3
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