Efﬁciency of International Standard Serial Number Code as an Error Correcting Scheme

: Error correcting coding is an effective technique of detecting and correcting errors which may occur due to environmental interference or physical defects such as human errors in the communication channels. The International Standard Serial Number code is internationally used for identifying the title of serial publications. This paper analyzes the efﬁciency of the international standard serial number code as an error correcting scheme. Moreover, the paper explores on the factors which affect the efﬁciency of any error correcting scheme. The study utilizes weight checksum technique to detect and correct error(s) in a code word. It is clear that ISSN code is not an efﬁcient error coding scheme. ISSN code is only reliable in error detection. ISSN code can detect any error in the code iff the weight checksum equation does not hold. However, the code does not detect silent errors. The study develops a new efﬁcient and robust modiﬁed ISSN code that is efﬁcient in error detection and correction capabilities. The code has dual mechanism for error detection and correction in a code word. If the weight checksum equation does not hold and secondly, if the conditions for the generating equation do not hold. Modiﬁed ISSN code can detect and correct silent errors in a code word. Modiﬁed ISSN code is an efﬁcient error coding scheme for it is efﬁcient in error detection and correction capabilities.


Introduction
Different communication channels have different error correcting coding schemes depending on the types of errors expected in a particular communication channel. An effective error coding requires an efficient scheme which is selected based on the characteristics of the specific communication channel.

Factors Affecting the Efficiency of an Error
Coding Scheme 1.1.1. Durability of the Entire Error Coding Scheme Durability of error coding scheme depends on the duration of time the entire coding system can stay before being exhausted. An efficient error coding scheme should last for a long period of time. The dictionary of the coding scheme should be large enough to be used for a longer period in its utilization.

Precision of Code Words in an Error Coding Scheme
The total number of digits in a code word in an efficient coding scheme should not be too large so as to lose its immediate utilization. A good (n, q) error correcting code should have a considerable length (n) for fast transmission of messages and large field q to ensure wide transmission of a variety of information ( to ensure big dictionary). The code words of the coding scheme should have little similarities to each other. A good (n, q) error correcting code should have a well define field q to ensure precision in its usage. The length of all code words should be same.

Reliability in Utilization of the Error Coding
Scheme The primary principle of an error coding scheme is to assist the receiver of a message to get the true information intended by the sender, therefore, an efficient error coding scheme should guarantee high probability of accuracy of the messages received by the receiver. An error coding scheme can only guarantee a high probability of accuracy of the message if and only if it can be able to detect and correct error(s) which may occur. An efficient error coding scheme should, therefore, detect and correct all errors that may occur from the original message.

Durability of the ISSN Code
The International Standard Serial Number (ISSN) is a code that is internationally accepted for identifying the title of serial publications. From ISSN network statistics by January 2019, more than 2.5 million ISSNs had been issued. The number increases by approximately 60, 000 to 70, 000 ISSNs annually. However, about 130, 000 ISSNs are changed and corrected annually. ISSN is only associated with the title of the publication. Therefore in case a publication is modified appreciably, a new ISSN has to be assigned. This means that if a publication title changes ten times the same publication has to be assigned ten new ISSN. It is clearly evident that the number of ISSNs changed and corrected is about double the number of ISSN increase per year.
Additionally, the entire ISSN coding scheme has a relatively small dictionary. The dictionary of 10, 000, 000 code words and currently more than 2.5 million ISSNs has been issued by January 2019. This implies that with about 70,000 code words issued per year and 130, 000 ISSNs changes and correction annually, the remaining ISSN code words can last for maximum of 35 years. With high number of publications increase, currently ISSN coding scheme is not durable.

Precision of the ISSN Code Words
ISSN code is computed modulo 11, therefore the code is a finite field. ISSN code being a finite field of order 11, each element of the code word is supposed to be chosen from the set F 11 = {0, 1, 2, ..., 9, 10} but the element 10 has two digits 1 and 0 , therefore, not applicable as an element of the code word. However, in the case when the check digit is 10, an alphabetic letter is used. An alphabetic letter is used to ensure that the length of the ISSN codes remains eight. The usage of the alphabetic letter only in the check digit but not is an element of the entire code word makes the ISSN code to loss its brevity and precision. ISSN uses numeric digits except in check digit where alphabetic digit is used. Therefore, there is a lot of similarities of the elements from one code word to another.

Reliability in the Utilization of the ISSN Code Words
From the properties of ISSN code [12], it is clear that the following deductions holds; 5. ISSN cannot detect and correct silent error(s) in a code word.
6. ISSN code cannot correct jump transpose error, double error, jump twin error, phonetic error, twin error, omission, and insertion error in a code word.
Corollary 1.1. ISSN code is not an efficient error coding scheme.
Proof. It is clear that ISSN code is neither durable nor precise error correcting scheme. Moreover, it is clear that ISSN is not reliable in error correction for it only corrects a single error and a transposition error in a code word. Therefore, ISSN code is not efficient in error correction.

Modified ISSN Code
The Modified ISSN code is developed and generated by the use of functions, permutations and the properties of the existing ISSN code. The code is developed on the basis of a good communication channel and an efficient error coding scheme. The main reason of using alphabetical digits is to replace two digits numerical digits hence avoiding confusing and enhancing preciseness. 3. The elements in each code word is generated by the generating equation defined by x i+1 = 3x i + 5 (mod 31), i = 1, 2, ..., 6 except the last digit of the code word. The last digit of code word is the check digit hence is computed. The first term(digit) of the generating equation is chosen randomly from the F 31 ∼ = Z 31 . 4. It is not a must for the generating equation defined by

Development and generation of modified ISSN
x i+1 = 3x i + 5 (mod 31), i = 1, 2, ..., 6 to start at the beginning of a code word. The sequence generated by of the generating equation can also be terminated in a code word. In the case where the sequence is terminated, digit 0 is inserted after the last digit of the terminated sequence and before the first digit of the other generating equation . The first term(digit) of the other generating equation is also chosen randomly from the F 31 ∼ = Z 31 . 5. The digit 0 does not have to necessarily be used when a sequence generated by the generating equation has been terminated, it may be used as an element of the code word itself. That is, digit 0 may be generated by the generating equation or may be at the beginning of the generating equation. In this case, digit 0 acts as a neutral digit. This implies that the generated sequence does not necessarily terminate once there is a digit 0. 6. In the case where the generating equation does not start at the beginning of a code word, the first digit of sequence is repeated from the beginning of a code word until where the start digit of the generating equation starts. Moreover, it is not a must for the generating equation to start at the beginning immediately after digit 0 in case the generated sequence had been terminated.
In the case where the generating equation does not start after digit 0, the first digit of the new sequence is repeated after digit 0 until where the start digit of generating equation starts. 7. ISSN codes are only associated with the title of a publication, in case changes and corrections are made to the title of a publication, the code word will have an extra digit called a blind digit. A blind digit is separated from the check digit by a hyphen. A blind digit does not affect the weight check sum of the entire code word. It is given depending on the number of changes made in the title of a publication. Importance of the blind digit is to help to know the number of times the changes and corrections have been made to a title of a publication. 8. The eight digits of the modified ISSN code must satisfy Examples of modified ISSN code; Proof. The Modified ISSN code is finite field of order 31 where each digit is chosen from the set F 31 ∼ = Z 31 . Each code word of the modified ISSN code has a length of 8 symbols,where each symbol is chosen from the set F 31 . Moreover, the eighth digit of the modified ISSN code is the check digit which is computed on the basis of weight checksum equation.The elements in each code word is generated by the generating equation defined by x i+1 = 3x i + 5 (mod 31) i = 1, 2, 3, ..., 6. It is not a must for the generating formula for the generation of the elements of a code word to start at the begin of the code word. Total numbers of the code words depends on the generating equation and digit 0 for the termination of the generated sequence. Digit 0 for the termination of the generating equation can occur from the second digit of a code word to the seventh digit of a code word. There are 31 ways of choosing the first digit of a code word, two ways of choosing the second digit, that is, through generating equation or repetition if generating equation does not start in the beginning of a code word. From the third digit to the seventh digit there are three ways of choosing each digit. There is only one way of choosing the eighth digit for it is computed to satisfy the weight checksum equation. Moreover, if there is a digit 0 in second digit, there are 31 ways of choosing the third digit, similar to the fourth digit until the seventh digit. Therefore there are 31 × 2 × 3 5 or 31 5 ways of choosing a code word, hence the dictionary of the modified ISSN code is 15066 + 28, 629, 151 = 28, 644, 217.

Calculation of the Check Digit in a Modified ISSN Code Word
Let the code word for modified ISSN be X = x 1 , x 2 , ..., x 7 without the check digit.
To compute the check digit, x 8 , since the code word digits x 1 , x 2 , ..., x 7 are known, . Now since 0 is additive identity of Z n , therefore x 8 is the additive inverse of ξ(mod 31). Thus x 8 ≡ −ξ(mod 31). Example 2.1. Calculate x 8 for the ISSN code word To compute x 8 , calculate ξ =

Error Detection in Modified ISSN Code
Modified ISSN code is an improved error coding scheme in term of error detection and correction capabilities. Let X = x 1 , x 2 , ..., x 8 be the code word for modified ISSN, then the weight checksum is computed  Proof. Suppose X = x 1 , x 2 , ..., x 8 is the modified ISSN code word and Y = x 1 , x 2 , ..., x τ −1 , y τ , x τ +1 , ..., x 8 with y τ = x τ + α α = 0 is the modified ISSN code word with a single error that has occurred in digit x τ , 1 ≤ τ ≥ 8. Then
Proposition 3.2. The Modified ISSN code detects silent error in a code word.
Proof. Suppose a single error has occurred in a modified ISSN code word but .., x τ −1 , y τ , x τ +1 , ..., x 8 with y τ = x τ +α α = 0 is the modified ISSN code word with a single error that has occurred in the position x τ then This implies that either 9 − τ or α is a multiple of 31 or 0. Since 9 − τ cannot be a multiple of 31or 0 for 1 ≤ τ ≤ 8 hence α = 0 and no error. However, there is a single error in the code word then there exist a digit of the code word x i = (3x i−1 + 5) (mod 31). Therefore x τ = (3x τ −1 + 5) (mod 31) and hence the silent error detected. Conversely, Suppose a single error has occurred in modified ISSN code but x i = (3x i−1 + 5) (mod 31) for i = 2, 3, .., 7 then Proof. Suppose X = x 1 , x 2 , ..., x 8 is the modified ISSN code word and Y = x 1 , x 2 , ..., x τ −1 , y τ , x τ +1 , ..., x 8 with y τ = x τ + α α = 0 is the modified ISSN code word with a single error that has occurred in digit y τ , 1 ≤ τ ≥ 8. The most important thing is the code to detect the position of the digit with error. From second digit to the second last digit of the code word is generated by the generating equation x τ +1 = 3x τ + 5 (mod 31) unless there is a repetition of digits or digit 0 indicating termination of the generating equation. By use of the generating equation x τ +1 = 3x τ + 5 (mod 31) the digit y τ = x τ + α α = 0 is detected by y τ = (3x τ −1 + 5) (mod 31). After the position of y τ is detected, the error is corrected by the computing y τ = (3x τ −1 + 5) (mod 31) and y τ = ( xτ+1−5 3 ) (mod 31) . In case x τ +1 < 5 or (x τ +1 − 5) is not divisible by 3 then Then the weight checksum equation is tested whether it holds. Additionally, when the error has been detected and the position of the error is known. Then the error can be corrected by Hence the error is corrected. Remark 3.2. The Modified ISSN is more efficient in single error detection and correction than ISSN code.

Transposition Error Detection and Correction in a Modified ISSN Code Word
Proposition 3.4. The Modified ISSN code detects all transposition error in a code word.
Proof. From the above prove (detection of transpose error) it is clear that x i = (3x i−1 + 5) (mod 31) i = 2, 3, ..., 7 x β = (3x τ + 5) (mod 31) Thus implying there is a transpose error which can be corrected by exchanging the digits x τ and x β . Remark 3.3. The Modified ISSN is more efficient in transpose error detection and correction than ISSN code.

Jump Transposition Error Detection and Correction in Modified ISSN Code
Proposition 3.5. The Modified ISSN code detects all jump transposition error in a code word.
Corollary 3.2. The Modified ISSN code corrects all jump transposition error in a code word.

Double Error Detection and Correction in Modified ISSN Code
Proposition 3.6. The Modified ISSN code detects and corrects all double error in the code word.

Efficiency of the Modified ISSN in Error Detection and Correction
Proposition 4.1. Modified ISSN code is an efficient error coding scheme.
Proof. Without loss of generality, modified ISSN code is efficient in error detection and correction. Moreover, modified ISSN has a blind digit which shows the number of corrections and modification done of the title of publication. ISSN coding scheme is only associated with the title of the publication. Therefore in case a publication is modified appreciably, only the blind digit changes. This means that if a publication is modified ten times the modified ISSN code shows that the title of the publication has being changed ten times. Moreover, modified ISSN code is a finite field where its elements are precisely selected and therefore an efficient coding scheme.

Conclusion
The study has shown that ISSN code is not an efficient error coding scheme. ISSN code is only efficient in error detection for it only reliable in error detection. ISSN code can detect any error in the code iff the weight checksum equation does not hold. However, the code does not detect silent errors. Modified ISSN code is efficient in error detection and correction capabilities. The code has dual mechanism of detection of errors in a code word. First, if the weight checksum equation does hold and secondly, if the generating equation does not hold. Modified ISSN code can detect and correct silent error in a code word. Modified ISSN code is an efficient error coding scheme for it is efficient in error detection and correction capabilities. Moreover, the code has a relatively big dictionar.