As described in the title , Data lossless compression limits from 1948 Shannon proposed “ Information entropy ” Since then, the theoretical limit of source coding has been defined . this , Based on my weighted probability model , I'm going to give you experiments that you can enumerate infinitely . The purpose of the experiment is to achieve the goal “ Lossless entropy reduction ” and “ Lossless entropy increasing ”. Those who have studied information theory and coding technology , Do you think I'm in the Arabian Nights ?

No hurry !! I'll start with three sets of data , Each group 256 individual （0-255 Random number of ） byte , Please do lossless compression for the following three groups of data ：

first group ：3,172,216,20,41,187,55,239,218,71,179,55,215,130,144,0,236,228,150,29,68,94,2,4,208,108,219,77,236,55,15,56,34,113,29,140,207,18,164,20,0,25,0,57,55,80,2,212,244,151,73,68,107,224,121,123,48,251,29,61,149,122,115,124,104,63,61,168,102,8,40,209,183,146,27,15,248,5,204,215,85,251,187,169,61,174,59,182,118,55,57,178,8,56,88,162,20,151,114,125,204,34,24,47,120,54,145,49,173,40,69,24,14,190,104,1,56,0,2,4,203,106,215,39,178,62,238,48,16,219,180,182,150,254,54,192,48,146,192,136,236,211,195,213,53,66,31,211,132,107,82,93,30,231,25,102,137,242,171,192,188,76,158,114,0,0,9,121,129,97,154,109,53,76,91,50,196,223,226,0,45,44,172,56,54,176,65,237,70,112,0,0,255,251,13,22,229,133,121,141,90,6,3,136,231,159,204,126,104,13,143,156,49,174,222,0,31,229,201,142,141,155,94,163,198,128,215,98,250,187,155,219,70,32,107,124,31,70,98,192,14,255,106,2,213,200,187,12,0,122,242,22,192,97,134,54,10,0,0,0

The second group ：7,46,42,168,175,245,71,130,223,45,146,35,246,16,38,52,173,51,88,77,167,52,159,151,27,152,28,136,53,212,88,28,55,237,186,182,57,199,175,107,61,165,19,205,7,180,107,136,230,217,18,17,95,82,253,96,96,224,116,35,233,22,226,29,126,211,121,151,124,49,21,10,22,153,1,138,61,43,6,104,114,152,230,58,106,252,11,91,189,16,224,123,196,156,254,160,219,158,213,185,35,59,145,65,221,160,249,180,16,182,185,174,8,253,211,177,57,59,221,87,222,188,52,193,167,12,82,218,228,93,35,118,64,157,112,175,182,213,54,207,254,87,214,187,154,134,7,15,134,218,237,64,63,214,55,23,86,118,180,27,107,82,142,75,20,239,65,217,244,132,58,167,95,138,245,9,223,51,78,192,139,130,63,0,239,64,147,152,103,212,42,232,169,197,211,26,138,85,43,67,1,118,206,242,68,49,38,176,85,133,195,173,242,1,126,87,61,215,137,159,252,185,150,174,137,200,73,189,232,182,99,244,162,98,121,198,241,21,205,150,137,112,141,101,105,113,77,5,228,85,23,139,252,79,63,166,197,152,0,0

The third group ：16,168,14,177,241,9,215,81,64,222,107,65,220,124,9,47,41,47,144,197,3,204,109,19,255,143,25,141,166,66,182,162,96,77,38,133,140,84,146,229,8,63,225,225,40,167,124,224,98,97,59,221,215,65,222,236,107,220,3,219,149,124,196,96,91,245,107,136,181,235,192,93,45,121,9,92,24,123,80,153,187,216,190,171,91,39,106,215,232,63,210,32,137,162,146,148,63,232,149,63,141,69,136,102,186,139,19,102,114,116,227,109,36,53,1,219,225,116,242,168,229,68,50,81,33,8,181,229,168,12,14,231,214,70,100,37,66,60,96,6,249,230,87,100,150,30,242,230,12,92,39,217,110,42,11,163,21,53,127,4,152,236,134,131,115,250,196,98,60,209,22,122,117,107,15,66,17,82,144,243,143,46,248,85,152,40,59,203,168,62,167,13,189,129,169,44,157,118,163,122,222,131,23,203,81,194,71,96,21,237,110,150,158,211,244,25,164,74,15,31,186,50,234,117,132,63,158,27,65,255,117,187,21,11,0,251,221,69,102,29,245,125,204,118,21,234,97,186,185,158,54,132,50,181,211,1,82,107,76,0

According to statistics , It's easy to draw the conclusion of the above three sets of data 1 The probabilities are 0.462981,0.504327,0.491346. According to information theory , It is almost impossible to find the rule in such random numbers for lossless compression .

According to information entropy , The above three sets of data will no longer be compressed （ I don't think it's necessary for me to describe the related theorem ）.

However, based on the weighted probability model proposed by me, all of them can be compressed after lossless entropy reduction . After entropy reduction , The first group can be losslessly compressed to 32 Bytes , The second group can be losslessly compressed to 64 Bytes , The third group can be losslessly compressed to 128 Bytes .

See here , Is it sniffing , How is that possible? ? The point of doing research is often you “ think ” impossible !

Next, I'll give you three sets of data ：

The third A group ：1,0,0,0,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0,0,0,1,0,0,1,1,0,0,1,0,0,0,1,0,0,1,0,0,1,0,0,1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,1,1,0,1,1,1,1,1,0,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,0,1,1,0,0,0,0,0,1,0,1,1,1,1,0,0,1,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,1,0,1,1,0,1,0,0,1,0,0,0,0,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0,0,1,0,0,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,0,0,0,0,1,1,0,0,1,0,0,0,1,1,1,0,0

The third B group ：1,3,2,0,1,0,0,3,0,0,2,3,1,3,0,1,2,2,1,2,1,2,2,3,3,3,1,2,2,2,1,3,3,3,0,2,2,3,3,3,2,1,3,3,3,0,0,1,2,0,2,2,2,0,1,0,0,3,1,1,2,2,2,2,3,2,2,1,2,1,0,0,0,3,0,3,2,1,1,1,0,1,1,2,3,1,3,3,1,0,1,2,2,2,1,0,1,2,1,0,1,2,2,2,1,3,1,0,3,3,3,0,0,3,3,2,2,0,1,3,2,3,2,2,0,3,1,1,1,0,0,0,3,2,3,2,3,0,0,0,0,1,3,2,2,3,0,2,1,2,3,0,3,3,3,0,3,2,3,2,3,0,1,3,1,1,3,2,1,1,2,2,2,1,1,3,2,1,3,3,3,3,1,2,0,2,2,0,3,1,3,2,0,2,0,3,1,2,0,1,3,1,2,2,2,1,0,0,1,0,1,1,0,3,1,2,1,2,0,2,0,0,0,2,2,1,2,0,0,3,3,0,2,1,2,1,2,3,1,0,1,2,1,2,0,2,2,3,2,1,3,3,2,1,2,0

The third C group ：3,9,6,7,7,13,11,13,6,14,8,4,6,15,4,7,6,3,2,14,1,5,5,13,7,6,6,14,13,7,3,1,4,10,10,0,8,4,7,8,10,6,10,5,0,12,13,12,11,5,4,15,9,5,5,13,13,7,10,8,12,7,5,11,8,15,3,12,2,0,4,14,12,3,11,10,13,0,1,15,8,0,14,7,15,14,10,12,4,3,1,11,1,2,6,7,9,0,9,2,14,8,11,8,3,15,1,12,5,3,8,6,6,9,9,13,10,8,13,2,6,0,2,13,5,15,14,13,13,5,3,10,2,6,4,12,9,0,7,1,0,13,11,11,12,11,12,9,2,11,5,8,10,7,6,8,1,7,6,3,14,5,10,12,7,0,6,5,5,14,3,4,6,6,2,6,1,13,0,3,15,14,11,5,8,15,4,4,11,2,5,3,3,10,5,1,6,15,10,8,15,3,14,10,15,5,5,13,15,6,13,4,9,11,7,1,8,2,5,7,13,8,5,5,3,10,13,0,5,11,7,15,15,5,15,13,3,3,10,10,13,15,10,3,8,8,0,4,2,6,1,11,13,1,11,15

obviously ,A Group according to 8 Bits are combined into one byte , So all you need to do is 32 Bytes ;B Group according to 2 Bits are combined into one byte , need 64 Bytes ;C Group according to 4 Bits are combined into one byte , need 128 Bytes . however A What is the relationship between group A and group 1 ?

Because I use the weighted probability model A The first group of data obtained by lossless entropy increasing , So the first set of data can be reduced to entropy A. That is to say, I can compress the first set of data losslessly to 32 Bytes , Lossless compression 8 times . In the same way , This shows that the second and third groups can be lossless compression .

I can tell you clearly , No matter how exhaustive the data of different lengths , As long as the random data can be obtained by lossless entropy increase , It can be reduced by lossless entropy . Here is a part of my experimental code ：
#include <stdio.h> #include <stdlib.h> #include <windows.h> #include <time.h>
#include "WJLEntropyTransformation.h" #ifdef WIN32 #define inline __inline
#endif // WIN32 // Symbols in each byte 1 The number of , Convenient statistics unsigned char CntOfOneSymboltemp[256]= {
0x00,0x01,0x01,0x02,0x01,0x02,0x02,0x03,0x01,0x02,0x02,0x03,0x02,0x03,0x03,0x04,
0x01,0x02,0x02,0x03,0x02,0x03,0x03,0x04,0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,
0x01,0x02,0x02,0x03,0x02,0x03,0x03,0x04,0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,
0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,
0x01,0x02,0x02,0x03,0x02,0x03,0x03,0x04,0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,
0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,
0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,
0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,0x04,0x05,0x05,0x06,0x05,0x06,0x06,0x07,
0x01,0x02,0x02,0x03,0x02,0x03,0x03,0x04,0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,
0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,
0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,
0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,0x04,0x05,0x05,0x06,0x05,0x06,0x06,0x07,
0x02,0x03,0x03,0x04,0x03,0x04,0x04,0x05,0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,
0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,0x04,0x05,0x05,0x06,0x05,0x06,0x06,0x07,
0x03,0x04,0x04,0x05,0x04,0x05,0x05,0x06,0x04,0x05,0x05,0x06,0x05,0x06,0x06,0x07,
0x04,0x05,0x05,0x06,0x05,0x06,0x06,0x07,0x05,0x06,0x06,0x07,0x06,0x07,0x07,0x08
}; int main(){ long cont1 = 0, cont2 = 0; unsigned int enoutlength = 512,
length = 512; unsigned int deoutlength = 512; int i = 0; unsigned char
*TestArray; unsigned char *enoutArray; unsigned char *deoutArray; double
ProbabilityofOne = 0.0; // random datas printf("the length is %d for random
datas:\n",length); TestArray = (unsigned char *)malloc(sizeof(unsigned char) *
length); srand(time(0)); for(i = 0; i < length; ++i) { TestArray[i] = rand() %
4; // This place can be changed to rand() % 2;rand() % 3;rand() % 16 wait printf("%d,",(unsigned
char)TestArray[i]); } printf("\n");
printf("------------------------------------test Increase Entropy
------------------------------------\n"); enoutArray =
WJLLosslessIncreaseEntropyCoding(TestArray, length, &enoutlength,
&ProbabilityofOne); for(i = 0; i < enoutlength; ++i) { printf("%d,",(unsigned
char)enoutArray[i]); cont1 += CntOfOneSymboltemp[enoutArray[i]]; }
printf("\n"); printf(" Symbols in raw data 1 The probability of ：%f,",ProbabilityofOne); printf("\n");
printf(" Symbols in the result of entropy increasing coding 1 The probability of ：%f,",(double)cont1/ (enoutlength * 8.0)); printf("\n");
printf("\n"); printf("-------------------------------------test Drop Entropy
---------------------------------------\n"); deoutArray =
WJLLosslessDropEntropyCoding(enoutArray, enoutlength, &deoutlength,
&ProbabilityofOne); for(i = 0; i < deoutlength; ++i) { printf("%d,",(unsigned
char)deoutArray[i]); if(deoutArray[i] != TestArray[i]){ cont2 ++; } }
printf("\n The number of differences is ：%d\n",cont2); printf("\n"); printf("\n"); system("pause");
return 0; }
next , I'll give you a few more pictures of how the console works ：

To prove that , I designed the experiment again , Can the random number generated by random function reduce entropy losslessly . Experiments show that forced entropy reduction can be achieved , Some numerical errors appear when entropy is increased after entropy reduction , And the more serious the entropy reduction is , The more mistakes entropy makes .

meanwhile , There is also a very small probability that the current random number can be reduced entropy losslessly . Here's what's going on 512 An experiment on the formation of random numbers ：

66,67,227,79,149,215,197,12,114,61,210,103,32,58,11,103,88,44,167,126,145,64,110,98,129,229,99,24,224,129,165,81,111,98,151,250,57,247,63,15,24,64,236,176,124,131,28,118,104,65,156,7,189,104,241,232,187,202,247,147,197,206,200,202,100,101,223,42,143,64,126,3,42,65,76,103,204,47,67,129,197,173,201,40,96,109,29,180,101,161,245,17,226,178,102,18,43,53,3,7,32,27,73,145,174,39,59,181,153,166,73,48,120,90,118,10,1,183,186,110,134,82,165,185,192,150,70,209,204,184,75,184,115,238,105,97,174,220,255,194,105,79,246,42,136,46,235,213,40,174,144,190,39,196,79,180,230,224,49,176,77,216,255,102,16,34,167,26,49,71,224,181,199,148,17,24,253,19,111,176,93,186,102,74,79,225,59,41,220,250,239,86,184,124,102,56,128,32,201,100,64,80,40,182,56,220,100,161,224,240,76,195,41,67,6,189,6,144,177,65,43,74,72,109,20,140,201,87,74,79,151,233,226,223,29,239,70,144,181,110,56,172,200,55,19,177,54,189,84,185,248,36,90,58,4,154,106,239,112,234,246,22,219,78,31,220,9,133,120,22,205,38,13,49,44,52,164,108,191,242,66,68,89,249,241,48,234,135,193,143,163,26,13,223,91,62,255,47,50,163,86,215,116,114,103,102,192,225,98,220,105,6,215,26,28,232,121,150,195,220,33,83,170,15,22,17,224,95,139,193,225,111,105,59,113,250,79,51,203,103,215,105,251,158,28,33,137,27,121,211,87,64,147,35,204,240,153,19,50,89,202,120,93,17,56,171,142,101,204,128,151,238,139,124,63,101,23,66,231,183,121,79,110,92,29,231,83,229,30,79,25,213,197,61,240,178,76,141,145,92,231,53,106,157,5,123,251,0,210,102,128,231,1,114,208,216,202,135,133,24,46,44,93,123,130,249,224,122,43,74,67,134,167,59,141,156,205,208,248,26,85,115,154,209,154,164,68,91,159,236,171,174,80,18,197,242,11,146,118,144,95,75,108,140,20,120,72,20,107,225,254,89,166,83,126,157,203,77,148,141,166,252,12,210,13,105,27,227,104,242,250,29,112,247,212,21,54,213,226,45,103,24,213,101,207,236,116,116,29,213,134,24

Put the symbol 1 The probability of is set to 0.6 Entropy reduction coding is used , The data obtained after entropy reduction are as follows ：

102,200,233,4,247,255,255,207,159,139,255,252,64,211,255,57,253,126,255,255,252,63,253,196,247,95,255,255,239,244,58,222,41,135,240,175,40,212,231,255,233,243,175,155,22,254,54,117,191,189,121,43,247,84,80,255,255,207,182,180,223,60,220,255,53,140,55,170,150,243,230,237,175,143,173,142,49,191,252,24,62,244,143,247,255,255,117,38,22,122,103,182,75,154,111,225,209,252,221,5,19,243,191,254,255,59,255,235,253,255,204,38,247,63,94,209,203,109,0,237,250,252,191,238,33,223,197,203,121,153,123,254,223,217,119,90,79,237,175,139,239,252,97,217,252,126,85,120,83,184,115,182,245,167,237,231,255,216,63,226,70,84,67,247,191,135,217,139,117,191,122,90,140,123,127,246,165,221,183,120,110,234,255,248,215,227,239,255,255,251,151,107,217,245,242,43,189,183,95,255,250,253,188,44,255,255,255,98,152,171,175,59,240,254,135,120,104,225,158,241,75,107,116,126,187,201,153,35,238,105,252,222,102,230,237,191,255,255,223,240,227,247,87,127,255,255,223,251,120,185,20,165,140,98,239,255,253,48,239,40,189,185,240,255,255,255,45,244,247,127,223,111,255,214,119,255,185,141,255,252,106,254,239,198,222,111,177,241,184,187,253,186,247,109,155,213,151,142,213,222,107,149,14,59,177,187,252,145,255,235,213,191,237,255,255,154,254,219,19,255,226,255,253,235,245,123,131,207,79,224,233,173,212,222,113,221,111,255,255,240,71,118,91,190,193,52,255,246,239,15,244,174,167,192,7,238,130,255,247,255,255,255,253,208,125,68,182,201,234,255,243,116,49,255,255,234,255,125,203,101,3,219,179,243,239,255,248,107,197,31,245,219,243,69,113,242,223,50,67,108,62,225,215,214,255,253,174,87,160,249,88,254,254,243,231,255,227,239,255,105,166,15,157,204,63,127,71,172,200,184,255,255,255,255,255,255,145,175,117,46,254,21,79,255,103,245,40,61,236,223,251,29,30,95,221,207,84,94,103,103,159,127,255,255,255,255,232,47,114,206,11,117,5,255,254,218,122,116,251,255,161,111,68,155,250,84,167,70,238,93,159,254,95,255,255,255,255,255,181,191,223,55,14,95,189,79,191,255,199,172,255,242

After statistics , Sign after actual entropy reduction 1 The probability is 0.680176, Needless to say, entropy reduction is successful . When entropy is increased, the following data are obtained ：

66,67,227,79,149,212***,165***,12,114,61,139***,103,32,58,11,101***,87***,43***,90***,126,134***,64,110,98,129,157***,96***,24,224,129,165,81,111,98,151,250,57,218***,63,15,24,64,236,176,124,131,28,118,104,65,156,7,189,95***,241,232,187,202,247,147,197,206,200,202,100,101,223,42,143,64,126,3,42,65,70***,103,204,47,67,128***,197,160***,201,40,96,109,29,180,101,161,245,17,226,177***,102,18,43,53,2***,231***,32,27,42***,145,174,39,59,181,153,165***,73,48,120,90,118,10,1,183,186,110,134,82,165,185,192,150,70,209,204,184,75,184,115,238,105,96***,174,220,255,194,105,79,246,41***,136,46,235,213,40,174,143***,190,39,196,79,180,230,224,49,176,77,216,255,101***,16,34,167,26,48***,71,224,181,199,148,17,24,207***,19,111,176,68***,169***,102,74,79,225,59,41,220,250,239,66***,184,124,102,56,127***,5***,201,100,64,80,40,182,55***,220,100,161,224,240,76,195,41,67,6,189,6,144,177,65,43,74,72,109,20,140,201,49***,3***,76***,150***,232***,226,222***,253***,219***,70,144,181,110,56,172,200,55,19,113***,54,189,84,185,248,36,90,45***,222***,154,106,239,110***,234,246,22,219,73***,31,219***,248***,133,120,22,205,38,13,49,44,52,164,108,191,242,66,68,89,249,241,48,234,135,193,143,163,26,13,223,91,61***,255,47,50,163,52***,215,116,114,103,85***,192,218***,98,220,105,6,215,26,28,232,121,150,195,220,33,83,125***,192***,22,17,224,95,139,193,225,102***,105,59,113,250,79,51,203,103,215,105,249***,155***,12***,245***,137,27,121,211,87,64,147,35,204,240,152***,242***,50,89,202,120,93,17,56,171,142,101,158***,128,151,238,137***,124,63,101,23,66,231,183,121,79,110,91***,29,231,83,211***,30,79,25,213,197,61,239***,178,76,135***,145,91***,231,53,106,157,5,123,251,0,210,102,128,230***,232***,26***,201***,167***,202,135,133,24,46,44,93,119***,130,249,224,122,43,74,67,134,167,57***,141,156,205,208,248,26,85,112***,100***,201***,154,164,68,91,159,236,171,174,80,18,197,242,11,145***,118,144,95,75,108,140,20,120,72,20,107,212***,254,27***,158***,2***,124***,157,202***,77,148,141,166,252,12,182***,13,105,24***,226***,227***,212***,50***

Among them are 90 Error in incrementing bytes （ As with *** The value of is wrong ）.

next , I said, why can this experiment shake “ information theory ”. At least I can find infinite kinds of random data that can be reduced by lossless entropy , So that means “ information theory ” There is incompleteness in the theory .

This time, I just recorded my experimental findings , I will provide it C Of lib Library for everyone to test .

in my opinion , This may be a new research direction , So let's send out the results first .

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