Author | Fitzpatrick, G. B. |
Source | ACM Digital Library |
Content type | Text |
Publisher | Association for Computing Machinery (ACM) |
File Format | |
Copyright Year | ©1960 |
Language | English |
Subject Domain (in DDC) | Computer science, information & general works ♦ Data processing & computer science |
Abstract | A number of papers have been written from time to time about logical counters of a certain type which have quite simple logic and have been variously referred to as Binary Ring Counters, Shift Register Counters, Johnson Counters, etc. To my knowledge, most of these papers confine themselves to certain special cases and usually leave the subject with some speculation as to the possibility of generating periods of any desired length by the use of these special types. The point of view of this paper is to consider all possible counters of this general type to see how one would obtain a particular period. Special emphasis is placed on determining the least number of bits, $\textit{n},$ required to produce a given period, $\textit{K}.The$ rules for counting are as follows. If an $\textit{n}-bit$ counter is in state $(\textit{a}\textit{n}-1,$ $\textit{a}\textit{n}-2$ ···, $\textit{a}2,$ $\textit{a}1,$ $\textit{a}0)$ at a given time, $\textit{T},$ then at $\textit{T}$ + 1 its state is $(\textit{b}\textit{n}-1,$ $\textit{b}\textit{n}-2,$ ···, $\textit{b}1,$ $\textit{b}0)$ where $\textit{b}0$ = $\textit{a}\textit{n}-1,$ $\textit{bi}$ = $\textit{a}\textit{i}-1$ + $\textit{ci}\textit{a}\textit{n}-1$ for $\textit{i}$ = 1, 2, ···, $\textit{n}$ - 1. The $\textit{a}'s,$ $\textit{b}'s,$ and $\textit{c}'s$ are all 0's or 1's, the $\textit{c}'s$ being constants, and the indicated operations are carried out using modulo 2 arithmetic. This is equivalent to considering the state of the counter as an $(\textit{n}$ - 1)th degree polynomial in $\textit{X},$ multiplying said polynomial by $\textit{X}$ and reducing it modulo $\textit{m}(\textit{X}),$ where $\textit{m}(\textit{X})$ is a polynomial of degree $\textit{n}$ which is relatively prime to $\textit{X}.$ At time $\textit{T}$ the state of the counter corresponds to: $\textit{A}(\textit{X})$ = $\textit{a}\textit{n}-1\textit{X}\textit{n}-1$ + $\textit{a}\textit{n}-2\textit{X}\textit{n}-2$ + ··· + $\textit{a}1\textit{X}$ + $\textit{a}0.$ The polynomial which corresponds to the state of the counter at time $\textit{T}$ + 1 is obtained by forming $\textit{X}·\textit{A}$ $(\textit{X})$ and reducing, if necessary, modulo $\textit{m}$ $(\textit{X})$ = $\textit{Xn}$ + $\textit{c}\textit{n}-1\textit{X}\textit{n}-1$ + $\textit{c}\textit{n}-2\textit{X}\textit{n}-2$ + ··· + $\textit{c}1\textit{X}$ + 1. Since $\textit{a}\textit{n}-1·\textit{m}(\textit{X})$ = 0 mod $\textit{m}(\textit{X}),$ $\textit{X}·\textit{A}(\textit{X})$ = $\textit{X}·\textit{A}(\textit{X})+$ $\textit{a}\textit{n}-1\textit{m}(\textit{X})$ mod $\textit{m}(\textit{X}),$ so $\textit{X}·\textit{A}(\textit{X})$ = $(\textit{a}\textit{n}-2$ + $\textit{c}\textit{n}-1·\textit{a}\textit{n}-1)\textit{X}\textit{n}-1$ + $(\textit{a}\textit{n}-3$ + $\textit{c}\textit{n}-2\textit{a}\textit{n}-1)\textit{X}\textit{n}-2$ + ··· + $(\textit{a}0$ + $\textit{c}1\textit{a}\textit{n}-1)\textit{X}$ + $\textit{a}\textit{n}-1$ = $\textit{b}\textit{n}-1\textit{X}\textit{n}-1$ + $\textit{b}\textit{n}-2\textit{X}\textit{n}-2$ + ··· + $\textit{b}1\textit{X}$ + $\textit{b}0.$ It is well known that more than one possible period may be obtained depending upon the initial state of the counter. Several examples are given by Young [4]. However, starting with $\textit{X}$ itself will always yield the longest possible period for any given $\textit{m}(\textit{X})$ and, furthermore, any other periods possible will always be divisors of the major period (Theorem I below). Since these minor periods can always be obtained with moduli of lower degree they are of no real interest here, and throughout the remainder of this paper the expression “period of the counter” will be assumed to refer to the major period.The set of all polynomials whose coefficients are the integers modulo 2 is the polynomial domain $\textit{GF}(2,$ $\textit{X}),$ which has among other things unique factorization into primes (irreducibles). If $\textit{m}(\textit{X})$ is in $\textit{GF}(2,$ $\textit{X}),$ then $\textit{GF}(2,$ $\textit{X})$ modulo $\textit{m}(\textit{X})$ is a commutative ring. Thus it is closed under multiplication, but it may have proper divisors of zero. However, any element which is relatively prime to $\textit{m}(\textit{X})$ in $\textit{GF}(2,$ $\textit{X})$ has an inverse in $\textit{GF}(2,$ $\textit{X})/\textit{m}(\textit{X})$ [1]. |
ISSN | 00045411 |
Age Range | 18 to 22 years ♦ above 22 year |
Educational Use | Research |
Education Level | UG and PG |
Learning Resource Type | Article |
Publisher Date | 1960-07-01 |
Publisher Place | New York |
e-ISSN | 1557735X |
Journal | Journal of the ACM (JACM) |
Volume Number | 7 |
Issue Number | 3 |
Page Count | 11 |
Starting Page | 287 |
Ending Page | 297 |
Ministry of Human Resource Development (MHRD) under its National Mission on Education through Information and Communication Technology (NMEICT) has initiated the National Digital Library of India (NDLI) project to develop a framework of virtual repository of learning resources with a single-window search facility. Filtered and federated searching is employed to facilitate focused searching so that learners can find out the right resource with least effort and in minimum time. NDLI is designed to hold content of any language and provides interface support for leading vernacular languages, (currently Hindi, Bengali and several other languages are available). It is designed to provide support for all academic levels including researchers and life-long learners, all disciplines, all popular forms of access devices and differently-abled learners. It is being developed to help students to prepare for entrance and competitive examinations, to enable people to learn and prepare from best practices from all over the world and to facilitate researchers to perform inter-linked exploration from multiple sources. It is being developed at Indian Institute of Technology Kharagpur.
NDLI is a conglomeration of freely available or institutionally contributed or donated or publisher managed contents. Almost all these contents are hosted and accessed from respective sources. The responsibility for authenticity, relevance, completeness, accuracy, reliability and suitability of these contents rests with the respective organization and NDLI has no responsibility or liability for these. Every effort is made to keep the NDLI portal up and running smoothly unless there are some unavoidable technical issues.
Ministry of Human Resource Development (MHRD), through its National Mission on Education through Information and Communication Technology (NMEICT), has sponsored and funded the National Digital Library of India (NDLI) project.
Phone: +91-3222-282435
For any issue or feedback, please write to ndl-support@iitkgp.ac.in