In fact this property that will be made use of in the best approximations. We just proceed on to give an example. These intuitive ideas are correct for some finite dimensional subspaces, but not for all infinite dimensional subspaces. We just enumerate some of the properties related with best approximation. Now this notion of best approximation for the first time is used in coding theory to find the best approximated sent code after receiving a message which is not in the set of codes used.
Further we use for coding theory only finite fields Fq. While defining the notion of inner product on vector spaces over finite fields we see all axiom of inner product defined over fields as reals or complex in general is not true. To overcome this problem we define for the first time the new notion of pseudo inner product in case of vector spaces defined over finite characteristic fields [9, ].
Then the pseudo inner product on. Let W be a subspace of V. Note: We need to see the difference even in defining our pseudo best approximation with the definition of the best approximation. Now having seen an example of the pseudo inner product we proceed on to illustrate by an example the notion of pseudo best approximation. Now having seen some simple properties of codes we now proceed on to define super special vector spaces. This new algebraic structure basically makes used of supermatrices. For more super linear algebra please refer [10, 21]. In this chapter we for the first time define a new class of super special vector spaces.
We describe mainly those properties essential for us to define super special codes and their properties like decoding, etc.
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Throughout this chapter V denotes a vector space over a field F. F may be a finite characteristic field or an infinite characteristic field. Vt a vector space of dimension m over F i. Example 2. Clearly Vs is a super special finite dimensional super vector space over Q. Clearly this Bs will generate Vs over Q. We have seen a super special finite dimensional vector space over Q.
Now we proceed on to define the super special basis of a super special vector space over F. If Bs forms a super linearly independent set and if every element in Vs can be expressed as a super linear combination from the set Bs in a unique way then we call Bs to be a special super basis of Vs or super special basis of Vs. If the number of elements in Bs is finite then we call Vs to be a finite dimensional super special vector space otherwise an infinite dimensional super special vector space. Suppose we consider a subset.
It is a matter of routine to verify a super special vector space which is finite dimensional will have only the same number of elements in every super special basis of Vs. Clearly both V1 and V2 are vector spaces of dimension 4 over Q. Ws is super special subspace of Vs. Then Vs is defined to be a super special mixed dimension vector space. We may have any other basis for Vs but it is true as in case of vector spaces even in special super vector spaces the number of elements in each and every superbasis is only fixed and equal.
Now we illustrate by an example the notion of super special mixed dimension vector space. V4 is a vector space of dimension 4 over Q. Thus Vs is a super special mixed dimensional vector space over Q. A super special subspace of Vs can be either a super special subspace or it can also be a super special mixed dimension subspace. Now it is important at this point to mention even a super special vector space can have a super special mixed dimension subspace also a super special mixed dimension vector space can have a super special vector subspace.
We illustrate this situation now by the following example. All the three vector spaces V1, V2 and V3 are of dimensions 6 over Q. Vs is a super special vector space over Q. W2 is a subspace of V2 of dimension 3. W3 is a subspace of V3 of dimension 3 over Q. Clearly Ws is a special super subvector space of varying dimension or Ws is a super special mixed dimension subspace of Vs but Vs is not a super special mixed dimension vector space, it is only a super special vector space over Q.
T3 is a subspace of V3 of dimension three over Q. Next we proceed on to give an example of a super special mixed dimension vector space having subspaces which are super special mixed dimension subspace and super special subspace. Clearly Ws is not a super special mixed dimensional subspace of Vs. R3 is a subspace of V3 of dimension 2 over Q. We see Rs is a super special mixed dimension subspace of Vs. We mention here only those factors about the super special vector spaces which are essential for the study and introduction of super special codes.
We will say a super special vector space endowed with super special inner product as a super special inner product space. While defining super special codes. We may need the notion of orthogonality. Also we see Vs is an abelian group with respect to addition. If Ws is a subspace of Vs it is necessarily a subgroup of Vs. When we are carrying this inner product to super special vector spaces over finite fields condition 4 may not in general be true.
It is easily verified. This property will also be used in super special codes. Since we are interested in only finite dimensional super special vector spaces that too defined over finite characteristic field we give some examples of them. Each of the spaces are of dimension 3 over Z2.
Clearly Vs is partitioned into 8 disjoint sets relative to the super special subspace Ws of Vs. Bs is a super special basis of the super special vector space Vs. Clearly Rs is a super special mixed dimension subspace of Vs. Ms is a super special basis of Rs. We see Rs is a subspace of dimension 4. Further we see all the super special base elements are only super row vectors.
We give yet another example of a super special mixed dimensional vector space over Z2. Clearly Rs is a super special subvector space of Ws and the super special dimension of Ws is 4. In the next chapter we define the notion of super special codes, which are built mainly using these super special vector spaces. In this chapter for the first time we define new classes of super special codes; using super matrices and enumerate some of their error correcting and error detecting techniques.
However their uses and applications would be given only in chapter four. This chapter has three sections. Section one introduces super special row codes and super special column codes are introduced in section two. Section three defines the new notion of super special codes and discusses their properties. In this section we define two new classes of super special row codes using super row matrix and super mixed row matrix.
For example. Clearly V is a super mixed row matrix or vector. For more refer chapter one of this book. That is the code word consisting of n code words can be represented as a super row vector;. These n code words denoted collectively by xs will be known as the super special row code word. Cs will also be known as the linear [ n1 n2 … nn , k1 k2 … kn ] or [ n1, k1 , n2, k2 , …, nn, kn ] super special row code. Clearly this will form the super special subspace of the super special vector space over Fq of super special dimension k1 k2 … kn. Cs being a super special subspace can be realized to be a group under addition known as the super special group code, where Hs is represented in the form given in equation I will be known as the standard form.
Example 3. The super special code words are given by. We define C1s to be the subcode of the super special rowcode Cs. C3s is a subcode of the super special row code Cs. Thus this code has several advantages which will be enumerated in the last chapter of this book. We give yet another example of super special code in which every super special code word is a super row vector and not a super mixed row vector. Now the super special system of equations is given by.
The subcode associated with super code Cs is. This gives T. So we can realize the super special row code to be codes in which each and every subcode Cis of Cs have the same number of check symbols. The super special code words associated with Cs are just super row vectors only and not super mixed row vectors.
The number of super special code words in Cs is 2n. Now having seen an example of a super special repetition row code we proceed on to define the super special parity check row code. We have two types of super special row parity check codes. Now we will illustrate the two types of super special parity check mixed row codes.
We see in every super row vector the number of non zero ones is even. Next we give an example of a super special parity check row code Cs. The super special codewords given by Hs is. Now having seen examples of the two types of super special parity check codes, we now proceed on to define super special Hamming distance, super special Hamming weight and super special errors in super special codes Cs. The super error vector is given by. As in case of usual linear codes we define super minimum distance d smin of a super special linear row code Cs as.
Thus the super minimum distance of Cs is equal to the least super weights of all non zero super special code words. Now the value of. Now d smin of the super special row code given in example 3. Hence we. We can define the new notion of super special syndrome to super code words of a super special row code which is analogous to syndrome of the usual codes. Thus this gives us a condition to find out whether the received super code word is a right message or not. We can find the correct word by the following method. Before we give this method we illustrate how the super special syndrome is calculated.
Now we have to shown how to find whether the received super special code word is correct or otherwise. With the advent of computers calculating the super special coset leaders is not a very tedious job. Appropriate programs will yield the result in no time. If every Csi is a cyclic code in Cs we call Cs to be a super special cyclic row code. We see each of C1s , Cs2 and C3s are cyclic codes. Now we see in general for any super special mixed row code with an associated super special parity check matrix Hs which happens to be a super mixed row matrix we cannot define the super special generator row matrix Gs.
So we shall first define the notion of super special generator row matrix of a super special row code mixed row code. The code words of C1s generated by G1 is given by. Clearly elements in Cs are super row vectors. Now we proceed to define super special mixed row code of Cs and its super special generator mixed row matrix. Clearly Gs is a super mixed row matrix. All the codes generated by G1, G2, G3 and G4 have the same number of message symbols.
We give a necessary and sufficient condition for Hs to have an associated generator matrix Gs for a super special row code Cs. We know every subcode Ci of Cs have the same number of check symbols. Hence the result. Thus we see in this situation we have both the super special generator row matrix of the code Cs as well as the super special parity check matrix of the code Cs are not super special mixed row matrices; we see both of them have the same length n for each row. We illustrate this situation by an example before we proceed on to define more concepts.
We see G1, G2 and G3 cannot be formed into a super mixed row matrix.
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This is in keeping with the theorem. Likewise if we have a super special row code Cs we may have the super special row matrix which generates Cs yet Hs may not exist. This is the marked difference between the super special row codes and usual linear codes. We see H1, H2 and H3 cannot be made into a super special row matrix. Hence the claim. Now having defined the new class of super special row mixed row codes we will now define new classes of mixed super classes of mixed super special row codes Cs i.
It is important to mention here that even if two types of classical codes are present still we call Cs as a mixed super special row code. Here C1 is a Hamming code, C2 the repetition code, C3 a code of no specific type and C4 a cyclic code. Let the mixed super special parity check matrix Hs associated with Cs be given by. Clearly Cs is a mixed super special mixed row code. C1 is the Hamming code, C2 any code and C3 a repetition code.
We define the new notion of super special Hamming row code. Then we call Cs to be a super special Hamming row code. The code words associated with C1s is. By taking one code word from C1s , one code word from Cs2 and one from C3s we form the super special Hamming row code. Suppose we are interested in finding super special codes in which the number of check symbols will be different for the subcodes.
In such a situation we see certainly we cannot work with the super special row codes Cs, for in this case we demand always the number of check symbols to be the same for every subcode in Cs, so we are forced to define this special or new classes of super special codes. Then we define it using super column matrix as the super code parity check matrix. We call Hs to be the super special parity check mixed column matrix of Cs and Cs is defined as the super special mixed column code.
The main difference between the super special row code and the super special column code is that in super special row codes always the number of check symbols in every code in Cs is the same as the number of message symbols in Ci and the length of the code Ci can vary where as in the super special column code, we will always have the same length for every code Ci in Cs but the number of message symbols and the number check symbols for each and every code Ci in Cs need not be the same.
In case if the number of check symbols in each and every code Ci is the same.
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Then we call Cs to be a super special column code. In case when we have varying number of check symbols then we call the code Cs to be a super special mixed column code. Thus every code is a n, k code. It may so happen that some of the Ci and Cj are identical codes. Now we proceed on to give examples of the two types of super special column codes.
Cm] would have the same length. It can have any number of message symbols and any arbitrary number of check symbols. Just we saw in example 3. But if we wish to have lesser transmission rate we cannot get it from these codes, the super special column codes comes handy. We now describe yet another super special column code Cs. We see every code word in Ci are of length 7. But the number of message symbols in C1 is 3 and the number of check symbols is 4 i. The code words in. The code.
Unlike in the case of super special row codes given a super special column code Cs with associated super special parity check column matrix Hs we will always be in a position to a get the super special generator matrix Gs. We see in the example 3. However for the example 3. We call Gs the super special generator column matrix which generates Cs. We say Gs is in the standard form only if each Gi is in the standard form. Like wise in the case of super special mixed column column code Cs if Gs is the super special column generator matrix in the standard form we can always get the super special column parity check matrix Hs from Hs and we have Gs H sT is always a super special zero column matrix.
As in case of super special row code we can define classical super special column codes. It is important to note that unlike the super special repetition row code which can have different lengths the super special repetition column code can have only a fixed length and any super special code word. Thus we see in case of super special column code it is impossible to get repetition codes of different lengths. However this is possible as seen earlier using super special row codes.
Similarly we can get only parity check super special codes to have same length. Let the super special parity check column matrix associated with Cs be given by. Thus we see we cannot get different lengths of parity check codes using the super special column code. However using super special row code we can get super special parity check codes of different lengths. We illustrate super special parity check column codes. The super special column parity check matrix.
Thus only when the user wants to send messages generated by the same parity check matrix he can use it. Now having seen this new class of codes using the parity check column code. We proceed on to built another new class of column codes using Hamming codes. Thus Cs has super special code words in it.
We can yet have different codes using different parity check matrices. So we can get different sets of codes and the number of elements in Cs is Now as in case of super special row codes we can in case of super special column codes have mixed super special column codes. The only criteria being is that each code Ci in Cs will be of same length. Now we proceed on to define them. The transmission rate of. We have given the basic definition and properties of super matrices in chapter one.
In this section we proceed on to define new classes of supper special codes and discuss a few properties about them. We call C S to be a super special code. H 1m ,H 2m , … ,H nm have the same number of rows. Array of super column vector messages. At the receiving end once again they are arranged back as the array of the column vectors which will be termed as the received message. Thus we can have transmission of two types at the source and the received message will accordingly be of two types viz. A typical super code word is a block. As an array of the super column vector we have.
While sending the array of super columns we send the message as. Now we would show how the received message is verified or found to be correct one or not by the following method. Though one can have other methods also to check whether the. Now we first describe the method using the general case before we proceed to explain with specific examples.
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To check whether the received super code word R S is correct or not, we make use of the super special syndrome technique. We can also use the method of best approximations and find the approximate sent message. We enumerate in this chapter a few applications and advantages of using the super special codes. Now the following will show why this super special super code is better than other usual code.
These codes can also be realised as a special type of concatenated codes. We enumerate them in the following:. Instead of using ARQ protocols we can use the same code C, in the super special super code. So that if the same message is sent from. We can take the correct message. This saves both money and time. We can also send it is as array of super row vectors i. In case the receiver wants to get the very correct message he can transmit both as an array of super row codes as well as, an array of super column codes the same code x and get the correct sent message.
Here it is not sending a message and receiving a message but is a study of transformation from time to time. It can be from satellite or pictures of heavenly bodies. Even in medical field this will find an immense use. Also these types of super special super codes can be used in scientific experiments so that changes can be recorded very minutely or with high sensitivity.
What one needs is a proper calibration linking these codes with those experiments were one is interested in observing the changes from time to time were the graphical representation is impossible due to more number of variables. Another striking advantage of the super special super code C S is that if one has to be doubly certain about the accuracy of the received message or one cannot request for second time transmission in those cases the sender can send super array of row codes and send the same codes as the super array of column codes.
After receiving both the messages, if both the received codes are identical and without any error one can accept it; otherwise find the error in each cell and accept the message which has less number of errors. If both the received messages have the same number of errors then if the machine which sends the code has a provision for a single or two or any desired number of cells alone can be non empty and other cells have empty code and send the message; in that case the super special code is taken as. At times the super special code C S may be of a special type i. The maximum number of super row vectors which happen to repeat would be.
If there is not even a single pair of coinciding super row vectors then we choose a super row which has a minimum number of errors. We find the correct code word from those cells in the ith row and accept it as the approximately correct received row. This form of transformation helps the receiver to study the real error pattern and the cells in C S which misbehave or that which always has an error message. Thus one can know not only more about the sent message but also know more about the problems in the machine while the message is transmitted; consequently corrections can be made so as to guarantee an approximately correct message is received.
The transmission can also take place in two ways simultaneously or by either array of super row transmission alone or array of super column transmission; only when we say the simultaneous transmission, we will first send the message x S in array of super row vector. If they are different for any row just by observation we can conclude the received message has an error and choose the row which has least number of differences. Now when array of column transmission takes place we see if y S is the received message then.
We take that column which has least number of errors or which has no error as the received message. The main advantage of simultaneous transmission is we can compare each cell of the received array of super row vectors and array of super column vectors. This type of code can be used when the ARQ process is impossible so that the same message can be filled in each cell of the super special code C S i.
That is if we have say for example a code C of length 8 with 16 code words only then we can choose C S to be a super special code with 16 rows and 17 columns. When these super special codes C S are used the user is certain to get marvellous results be it in any discipline. The class of super special super codes C S will be very beneficial in the cryptography for two main reasons. It is easy to be operated or transmitted and b. We just show. Very different codes which do not carry the message will also be repeated so that even by the frequency of the repetition intruder even cannot guess.
Only the concerned who are the reliable part and parcel of the communication work knows the exact super rows which carry the messages so they would only look into that super row to guarantee the error freeness during transmission. Several super rows carry the same message.
In our opinion this method with larger number of rows and columns in C S would make it impossible for the intruder to break it. We give yet another super special code C S , which is impossible for any intruder to break and which guarantees the maximum security. This is done for each and every row. Only the concerned persons who are part and parcel of the group know at what stages and which codes carry the message and such keys are present only with the group who exchange the information so it is impossible for any intruder to break it and high percentage of confidentiality and security is maintained.
Let C S be a super special super code. Suppose there are n codes arranged in the column and m codes along the rows of the super matrix of the code C S. The cryptographer can choose some blocks in C S to carry the messages and rest of the blocks may be used to mislead the intruder. When this type of super special codes are used it is impossible for any one to break and get into the structure. Now even in this block the cryptographer can use only certain rows and columns to carry true message and the rest only to mislead the intruder.
Here C is say a n, k code. The codes C is assumed to carry the true messages. Now this key will be known to every one in the group who is sending or receiving the messages. Thus any one in the group only will be interested in the rows 1, 4 and 5 and ignore the messages in the rows 2, 3 and 6. Thus when the number of rows and columns used are arbitrarily large it will be impossible for the intruder to guess at the codes and their by brake the key. Similarly one can use a. The group which uses this super special codes can agree upon to use the code C to carry the messages and C1 are misleading codes.
So anyone in this group will analyse only the codes in columns 2, 5 and 6 ignore columns 1, 3 and 4. We give now the example of a block and misleading block code for the cryptographist. Now any super code word x S in C S would be of the form. If y S is the received message. Now only few blocks are real blocks for which we have to work and other blocks are misleading blocks. Since all the codes have the same number of check symbols and the length of all the 12 codes are the same the intruder will not be in a position to make any form of guess and break the message.
In fact he will not even be in a position to find out which of the blocks are misleading block of C S and which of them really carries the message. Thus this provides a very high percentage of confidentiality and it is very difficult to know or break the message. This code when properly used will be a boon to the cryptography. We give yet another example of a super special code C S which would be of use to the cryptographist.
All the seven codes are of same length and same number of check symbols and message symbols. Here only C is true, all the other 6 codes C1, C2, …, C6 are only misleading codes. The super special parity check matrix H S is given by. Here only the parity check matrix H gives the needed message, all other parity check matrices H1, H2, H3, H4, H5 and H6 need not be even known to the owner of this system of cryptography.
This will be known as the true chart of C S. These types of super special super codes C S can be thought of as special steganography. Unlike in a stegnanography where a secret message would be hidden with other messages here only the secret message is the essential message and all other messages are sent as misleading messages so that the intruder is never in a position to break the key or get to know the message. Example 4. We call this to be a steganographic super special code.
For instance we have a group which works with some name which starts in T and the messages sent to one another is highly confidential for it involves huge amount of money transactions or military secrets. Now the super special code C S is given by. The hidden message is that the receiver is advised to use the truth table or a steganographic image of T in C S i.
For instance K is the first alphabet of some member of the group then we need to decode the message from. This will be the key he has to use to decode the received message. He needs to decode only the messages y1, y7, y13, y19, y25, y31, y37, y43, y49, y26, y21, y16, y11, y6, y33, y40, y47 and y54 which is easily seen to form the letter K. For instance if y S is the received message given by. The receiver should and need to decode only y4 y11 y18 y25 y32 y38 y46 y15 y16 y17 y19 y15 y20 and y This forms the cross.
Thus these can also be given as finite series or a finite arithmetic progression for instance arithmetic progression with first term 4 and difference 7 last term 46 and arithmetic progression with first term 15 common difference 1 and last term Any nice mathematical technique can be used to denote the codes which is essential for the member to be decoded to get the message. Any super code word x S in C S is of the form. This type of super matrix from now on wards will be known as super row cell matrix i. Now x S , y S can also be called as super row cell vectors.
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Now we define the dot product of two super cell vectors if and only if they are of same order i. Basically all these row vectors are from the subspaces of the vector spaces. Now how does the dot or inner product of x S with y S look like. Now having defined the notion of inner product of super row cell vectors or matrices, we proceed to define super orthogonal super code.
Suppose there exists a super code word y S in C S where. Now having defined cyclic orthogonal super special super codes C S we just indicate how error is detected and corrected. New crime index for buyers. Garden-variety roses just got an electrical upgrade. Roses rigged with electrical circuits.
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