DOCSLIB.ORG

Digital Circuit and Logic Design I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Digital Circuit and Logic Design I Digital Circuit And Logic Design I Lecture 2 Outline Number System and Codes (2) 1. Codes 2. Error detecting and correcting Panupong Sornkhom, 2005/2 2 Number System and Codes (2) 1. Codes What is code Decimal number codes Gray code Character codes Code for action, condition and states Panupong Sornkhom, 2005/2 4 What is code? A digital system requires that all information be in binary form. The external world, however, uses various other symbols to represent information. A symbol is represented in a digital system by a string of bits A set of n-bit strings in which different bit strings represent different symbol is called a code. A particular pattern of string is called code word. Panupong Sornkhom, 2005/2 5 Decimal number codes Binary-Coded Decimal Excess 3 code (BCD) code Self-complementing but is Encode the digit 0 – 9 by not weighted code their 4-bit unsigned binary Excess3 = BCD + 00112 representation, other code Biquinary code word are not used Uses 7 bits per code word BCD is a weighted code, the weight are 8,4,2,1. so, The first two bits indicate sometime this code called range of number 0-4 or 5-9, 8421 code. and the last five bits indicate which of the five 2421 code numbers in the selected Weighted code with self- range is represented complementing (9’s 1-out-of-10 code complement) Using 10-bit code word. Panupong Sornkhom, 2005/2 6 Decimal codes Decimal BCD 2421 Excess-3 Biquinary 1-out-of-10 digit (8421) 0 0000 0000 0011 0100001 1000000000 1 0001 0001 0100 0100010 0100000000 2 0010 0010 0101 0100100 0010000000 3 0011 0011 0110 0101000 0001000000 4 0100 0100 0111 0110000 0000100000 5 0101 1011 1000 1000001 0000010000 6 0110 1100 1001 1000010 0000001000 7 0111 1101 1010 1000100 0000000100 8 1000 1110 1011 1001000 0000000010 9 1001 1111 1111 1010000 0000000001 Panupong Sornkhom, 2005/2 7 Gray code It is sometimes necessary for an input sensor to produce a digital value that indicates a mechanical position. The encoder has a problem when the disk is positioned at certain boundaries between the regions where more than one bit 3-bit binary code changes This problem can be solved by devising a digital code in which only one bit changes between each pair of successive code words. Such a code is called Gray code 3-bit Gray code Panupong Sornkhom, 2005/2 8 Gray code (cont.) There are two convenient ways to Decimal 1 bit 2 bits 3 bits construct n-bit Gray code The first method is based on the 0 0 00 000 fact that Gray code is a reflected code 1 1 01 001 A 1-bit Gray code has two code words, 0 and 1. 2 - 11 011 The first 2n code words of an (n+1)-bit Gray code equal the 3 - 10 010 code words of an n-bit Gray code, written in order with a 4 - - 110 leading 0 append The last 2n code words of an 5 - - 111 (n+1)-bit Gray code equal the code words of an n-bit Gray 6 - - 101 code, written in reverse order with a leading 1 append 7 - - 100 Panupong Sornkhom, 2005/2 9 Gray code (cont.) The second method allows us Decimal Binary code Gray code to derive n-bit Gray-code code number word directly from the 0 000 000 corresponding n-bit binary code word 1 001 001 The bits of an n-bit binary or 2 010 011 Gray-code code word are numbered from right to left, 3 011 010 from 0 to n-1 Bit i of a Gray-code code word 4 100 110 is 0 if bits i and i+1 of the corresponding binary code 5 101 111 word are the same, else bit i is 1. (When i+1 = n, bit n of the 6 110 101 binary code word is considered to be 0.) 7 111 100 Panupong Sornkhom, 2005/2 10 Character codes The most commonly used character code is ASCII, the American Standard Code for Information Interchange. ASCII represents each character with a 8-bit string, in fact 7 bits are sufficient for English The code contains the uppercase and lowercase alphabet, numerals, punctuation, and various nonprinting control characters. Panupong Sornkhom, 2005/2 11 Character codes (cont.) Pictures from http://www.jimprice.com/jim-asc.htm Panupong Sornkhom, 2005/2 12 Character codes (cont.) Pictures from http://www.jimprice.com/jim-asc.htm Panupong Sornkhom, 2005/2 13 Codes for Actions, Conditions, and States Characters and numbers are considered to be data but sometimes we need non-data codes to represent actions, conditions, or states. For example, consider a simple traffic-light controller. State Lights N-S Green N-S Yellow N-S E-W Green E-W Yellow E-W Code word Red Red N-S go ON OFF OFF OFF OFF ON 000 N-S wait OFF ON OFF OFF OFF ON 001 N-S delay OFF OFF ON OFF OFF ON 010 E-W go OFF OFF ON ON OFF OFF 100 E-W wait OFF OFF ON OFF ON OFF 101 E-W delay OFF OFF ON OFF OFF ON 110 Panupong Sornkhom, 2005/2 14 2. Error detecting and correcting Introduction Single error detecting Error correcting and multiple errors detecting Hamming code Panupong Sornkhom, 2005/2 15 Introduction An error in a digital system is the corruption of the data from its correct value to some other value caused by physical failure Failure can be either temporary or permanent Single error = failure affect only a single bit of data Multiple errors = two or more bits in error Error-detecting codes n A code that uses n-bit strings need not contain 2 valid code words Corrupting a code word will likely produce a bit string that is not a code word Panupong Sornkhom, 2005/2 16 Single error detecting A code detects all single errors Information Even-parity Odd-parity is the minimum distance bits code code between all possible pairs of 000 000 0 000 1 code words is 2 001 001 1 001 0 In general, we need n+1 bits to 010 010 1 010 0 construct a single-error- detecting code with 2n code 011 011 0 011 1 words. 100 100 1 100 0 The first n bits of a code word = information bits 101 101 0 101 1 The last one bit = parity bit 110 110 0 110 1 111 111 1 111 0 Panupong Sornkhom, 2005/2 17 Error-correcting and multiple-error-detecting codes Assume that failures affect at most one bit of each code word, then a noncode word with a 1-bit error will be closer to the originally code word than to any other code word. Therefore, we can correct the error by changing the noncode word to the nearest code word A code that is used to correct errors is called an error- correcting code If a code has minimum distance 2c+1, it can be used to correct errors that affect up to c bits If a code’s minimum distance is 2c+d+1, it can be used to correct errors in put to c bits and to detect errors in put to d additional bits Panupong Sornkhom, 2005/2 18 Error-correcting and multiple-error- detecting codes (cont.) For example, a code with minimum distance 4 (c=1, d=1) Single-bit errors that produce noncode words 00101010 and 11010011 can be corrected However, a error that produces 10100011 can’t be corrected but can be detected Pictures from Textbook DDPP Panupong Sornkhom, 2005/2 19 Error-correcting and multiple-error-detecting codes (cont.) A 3-bit error may be “corrected” to the wrong value. It may be acceptable if 3- bit errors are rarely to occur. However, if we concerned about 3-bit errors, we can Picture from Textbook DDPP change the decoding policy for the code by just flag all noncode words as uncorrectable errors. Panupong Sornkhom, 2005/2 20 Hamming codes In 1950, R. W. Hamming described a general method for constructing codes with a minimum distance of 3, now called Hamming codes For any value of i, his method yields a (2i – 1)-bit code (check bits = i bits and information bits = 2i –1 –i bits) The bit positions can be numbered from 1 through 2i-1. Any position whose number is power of 2 is a check bit The remaining positions are information bits Panupong Sornkhom, 2005/2 21 Hamming codes (cont.) Each check bit is grouped with a subset of the information bits, as specified by a parity-check matrix. Each check bit is grouped with the information positions whose numbers have a 1 in the same bit when expressed in binary For example, check bit 2 (010) is grouped with information bits 3 (011), 6 (110), and 7 (111) For a given combination of information-bit values, each check bit is chosen to produce even parity, so the total number of 1s in its group is even. Panupong Sornkhom, 2005/2 22 Hamming codes (cont.) Parity-check matrices for 7-bit Hamming codes: (a) With bit positions in numerical order; (b) With check bits and information bits separated Panupong Sornkhom, 2005/2 23 Hamming codes (cont.) Example Distance-3 Hamming code Information bits = 0000 Î Parity bits = 000 Information bits = 0100 Î Parity bits = 110 Information bits = 0111 Î Parity bits = 000 Information bits = 1100 Î Parity bits = 001 Information bits = 1101 Î Parity bits = 010 A distance-3 Hamming code can easily be modified to increase its minimum distance to 4.
Load more

Recommended publications
  • Final-Del-Lab-Manual-2018-19
    LAB MANUAL DIGITAL ELECTRONICS LABORATORY Subject Code: 210246 2018 - 19 INDEX Batch : - Page Date of Date of Signature of Sr.No Title No Conduction Submission Staff GROUP - A Realize Full Adder and Subtractor using 1 a) Basic Gates and b) Universal Gates Design and implement Code converters-Binary to Gray and BCD to 2 Excess-3 Design of n-bit Carry Save Adder (CSA) and Carry Propagation Adder (CPA). Design and Realization of BCD 3 Adder using 4-bit Binary Adder (IC 7483). Realization of Boolean Expression for suitable combination logic using MUX 4 74151 / DMUX 74154 Verify the truth table of one bit and two bit comparators using logic gates and 5 comparator IC Design & Implement Parity Generator 6 using EX-OR. GROUP - B Flip Flop Conversion: Design and 7 Realization Design of Ripple Counter using suitable 8 Flip Flops a. Realization of 3 bit Up/Down Counter using MS JK Flip Flop / D-FF 9 b. Realization of Mod -N counter using ( 7490 and 74193 ) Design and Realization of Ring Counter 10 and Johnson Ring counter Design and implement Sequence 11 generator using JK flip-flop INDEX Batch : - Design and implement pseudo random 12 sequence generator. Design and implement Sequence 13 detector using JK flip-flop Design of ASM chart using MUX 14 controller Method. GROUP - C Design and Implementation of 15 Combinational Logic using PLAs. Design and simulation of - Full adder , 16 Flip flop, MUX using VHDL (Any 2) Use different modeling styles. Design & simulate asynchronous 3- bit 17 counter using VHDL. Design and Implementation of 18 Combinational Logic using PALs GROUP - D Study of Shift Registers ( SISO,SIPO, 19 PISO,PIPO ) Study of TTL Logic Family: Feature, 20 Characteristics and Comparison with CMOS Family Study of Microcontroller 8051 : 21 Features, Architecture and Programming Model ` Digital Electronics Lab (Pattern 2015) GROUP - A Digital Electronics Lab (Pattern 2015) Assignment: 1 Title: Adder and Subtractor Objective: 1.
    [Show full text]
  • A New Approach to the Snake-In-The-Box Problem
    462 ECAI 2012 Luc De Raedt et al. (Eds.) © 2012 The Author(s). This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-098-7-462 A New Approach to the Snake-In-The-Box Problem David Kinny1 Abstract. The “Snake-In-The-Box” problem, first described more Research on the SIB problem initially focused on applications in than 50 years ago, is a hard combinatorial search problem whose coding [14]. Coils are spread-k circuit codes for k =2, in which solutions have many practical applications. Until recently, techniques n-words k or more positions apart in the code differ in at least k based on Evolutionary Computation have been considered the state- bit positions [12]. (The well-known Gray codes are spread-1 circuit of-the-art for solving this deterministic maximization problem, and codes.) Longest snakes and coils provide the maximum number of held most significant records. This paper reviews the problem and code words for a given word size (i.e., hypercube dimension). prior solution techniques, then presents a new technique, based on A related application is in encoding schemes for analogue-to- Monte-Carlo Tree Search, which finds significantly better solutions digital converters including shaft (rotation) encoders. Longest SIB than prior techniques, is considerably faster, and requires no tuning. codes provide greatest resolution, and single bit errors are either recognised as such or map to an adjacent codeword causing an off- 1 INTRODUCTION by-one error.
    [Show full text]
  • NOTE on GRAY CODES for PERMUTATION LISTS Seymour
    PUBLICATIONS DE L’INSTITUT MATHEMATIQUE´ Nouvelle s´erie,tome 78(92) (2005), 87–92 NOTE ON GRAY CODES FOR PERMUTATION LISTS Seymour Lipschutz, Jie Gao, and Dianjun Wang Communicated by Slobodan Simi´c Abstract. Robert Sedgewick [5] lists various Gray codes for the permutations in Sn including the classical algorithm by Johnson and Trotter. Here we give an algorithm which constructs many families of Gray codes for Sn, which closely follows the construction of the Binary Reflexive Gray Code for the n-cube Qn. 1. Introduction There are (n!)! ways to list the n! permutations in Sn. Some such permutation lists have been generated by a computer, and some such permutation lists are Gray codes (where successive permutations differ by a transposition). One such famous Gray code for Sn is by Johnson [4] and Trotter [6]. In fact, each new permutation in the Johnson–Trotter (JT ) list for Sn is obtained from the preceding permutation by an adjacent transposition. Sedgewick [5] gave a survey of various Gray codes for Sn in 1977. Subsequently, Conway, Sloane and Wilks [1] gave a new Gray code CSW for Sn in 1989 while proving the existence of Gray codes for the reflection groups. Recently, Gao and Wang [2] gave simple algorithms, each with an efficient implementation, for two new permutation lists GW1 and GW2 for Sn, where the second such list is a Gray code for Sn. The four lists, JT , CSW , GW1 and GW2, for n = 4, are pictured in Figure 1. This paper gives an algorithm for constructing many families of Gray codes for Sn which uses an idea from the construction of the Binary Reflected Gray Code (BRGC) of the n-cube Qn.
    [Show full text]
  • Other Binary Codes (3A)
    Other Binary Codes (3A) Young Won Lim 1/3/14 Copyright (c) 2011-2013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to [email protected]. This document was produced by using OpenOffice and Octave. Young Won Lim 1/3/14 Coding A code is a rule for converting a piece of information (for example, a letter, word, phrase, or gesture) into another form or representation (one sign into another sign), not necessarily of the same type. In communications and information processing, encoding is the process by which information from a source is converted into symbols to be communicated. Decoding is the reverse process, converting these code symbols back into information understandable by a receiver. Young Won Lim Other Binary Codes (4A) 3 1/3/14 Character Coding ASCII code BCD (Binary Coded Decimal) definitions for 128 characters: Number characters (0-9) 33 non-printing control characters (many now obsolete) 95 printable charactersi Young Won Lim Other Binary Codes (4A) 4 1/3/14 Representation of Numbers Fixed Point Number representation ● 2's complement +1234 ● 1's complement 0 -582978 coding ● sign-magnitude Floating Point Number representation +23.84380 ● IEEE 754 -1.388E+08
    [Show full text]
  • Loopless Gray Code Enumeration and the Tower of Bucharest
    Loopless Gray Code Enumeration and the Tower of Bucharest Felix Herter1 and Günter Rote1 1 Institut für Informatik, Freie Universität Berlin Takustr. 9, 14195 Berlin, Germany [email protected], [email protected] Abstract We give new algorithms for generating all n-tuples over an alphabet of m letters, changing only one letter at a time (Gray codes). These algorithms are based on the connection with variations of the Towers of Hanoi game. Our algorithms are loopless, in the sense that the next change can be determined in a constant number of steps, and they can be implemented in hardware. We also give another family of loopless algorithms that is based on the idea of working ahead and saving the work in a buffer. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases Tower of Hanoi, Gray code, enumeration, loopless generation Contents 1 Introduction: the binary reflected Gray code and the Towers of Hanoi 2 1.1 The Gray code . .2 1.2 Loopless algorithms . .2 1.3 The Tower of Hanoi . .3 1.4 Connections between the Towers of Hanoi and Gray codes . .4 1.5 Loopless Tower of Hanoi and binary Gray code . .4 1.6 Overview . .4 2 Ternary Gray codes and the Towers of Bucharest 5 3 Gray codes with general radixes 7 4 Generating the m-ary Gray code with odd m 7 5 Generating the m-ary Gray code with even m 9 arXiv:1604.06707v1 [cs.DM] 22 Apr 2016 6 The Towers of Bucharest++ 9 7 Simulation 11 8 Working ahead 11 8.1 An alternative STEP procedure .
    [Show full text]
  • Algorithms for Generating Binary Reflected Gray Code Sequence: Time Efficient Approaches
    2009 International Conference on Future Computer and Communication Algorithms for Generating Binary Reflected Gray Code Sequence: Time Efficient Approaches Md. Mohsin Ali, Muhammad Nazrul Islam, and A. B. M. Foysal Dept. of Computer Science and Engineering Khulna University of Engineering & Technology Khulna – 9203, Bangladesh e-mail: [email protected], [email protected], and [email protected] Abstract—In this paper, we propose two algorithms for Coxeter groups [4], capanology (the study of bell-ringing) generating a complete n-bit binary reflected Gray code [5], continuous space-filling curves [6], classification of sequence. The first one is called Backtracking. It generates a Venn diagrams [7]. Today, Gray codes are widely used to complete n-bit binary reflected Gray code sequence by facilitate error correction in digital communications such as generating only a sub-tree instead of the complete tree. In this digital terrestrial television and some cable TV systems. approach, both the sequence and its reflection for n-bit is In this paper, we present the derivation and generated by concatenating a “0” and a “1” to the most implementation of two algorithms for generating the significant bit position of the n-1 bit result generated at each complete binary reflected Gray code sequence leaf node of the sub-tree. The second one is called MOptimal. It () ≤ ≤ n − is the modification of a Space and time optimal approach [8] G i ,0 i 2 1 , for a given number of bits n. We by considering the minimization of the total number of outer evaluate the performance of these algorithms and other and inner loop execution for this purpose.
    [Show full text]
  • GRAPHS INDUCED by GRAY CODES 1. Introduction an N-Bit
    GRAPHS INDUCED BY GRAY CODES ELIZABETH L. WILMER AND MICHAEL D. ERNST Abstract. We disprove a conjecture of Bultena and Ruskey [1], that all trees which are cyclic graphs of cyclic Gray codes have diameter 2 or 4, by producing codes whose cyclic graphs are trees of arbitrarily large diameter. We answer affirmatively two other questions from [1], showing that strongly Pn Pn- compatible codes exist and that it is possible for a cyclic code to induce× a cyclic digraph with no bidirectional edge. A major tool in these proofs is our introduction of supercomposite Gray codes; these generalize the standard reflected Gray code by allowing shifts. We find supercomposite Gray codes which induce large diameter trees, but also show that many trees are not induced by supercomposite Gray codes. We also find the first infinite family of connected graphs known not to be induced by any Gray code | trees of diameter 3 with no vertices of degree 2. 1. Introduction n An n-bit Gray code B = (b1; b2;:::; bN ), N = 2 , lists all the binary n-tuples (\codewords") so that consecutive n-tuples differ in one bit. In a cyclic code, the first and last n-tuples also differ in one bit. Gray codes can be viewed as Hamiltonian paths on the hypercube graph; cyclic codes correspond to Hamiltonian cycles. Two Gray codes are isomorphic when one is carried to the other by a hypercube isomorphism. The transition sequence τ(B) = (τ1; τ2; : : : ; τN 1) of an n-bit Gray code B lists − the bit positions τi [n] = 1; 2; : : : ; n where bi and bi+1 differ.
    [Show full text]
  • Lab Manual Exp Code Converter.Pdf
    Hybrid Electronics Laboratory Design and Simulation of Various Code Converters Aim: To Design and Simulate Binary to Gray, Gray to Binary , BCD to Excess 3, Excess 3 to BCD code converters. Objectives: 1. To understand different codes 2. To design various Code converters using logic gates 3. To simulate various code converters 4. To understand the importance of code converters in real life applications Theory Binary Codes A symbolic representation of data/ information is called code. The base or radix of the binary number is 2. Hence, it has two independent symbols. The symbols used are 0 and 1. A binary digit is called as a bit. A binary number consists of sequence of bits, each of which is either a 0 or 1. Each bit carries a weight based on its position relative to the binary point. The weight of each bit position is one power of 2 greater than the weight of the position to its immediate right. e. g. of binary number is 100011 which is equivalent to decimal number 35. BCD Codes Numeric codes represent numeric information i.e. only numbers as a series of 0’s and 1’s. Numeric codes used to represent decimal digits are called Binary Coded Decimal (BCD) codes. A BCD code is one, in which the digits of a decimal number are encoded-one at a time into group of four binary digits. There are a large number of BCD codes in order to represent decimal digits0, 1, 2 …9, it is necessary to use a sequence of at least four binary digits.
    [Show full text]
  • A Review of Binary to Gray and Gray to Binary Code Conversion
    Volume 3, Issue 10, October – 2018 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 A Review of Binary to Gray and Gray to Binary Code Conversion Pratibha Soni Vimal Kumar Agarwal M. TECH. Scholar Associate Professor Apex Institute of Engineering and Technology Apex Institute of Engineering and Technology Jaipur, Rajasthan, India Jaipur, Rajasthan, India Abstract:- In today’s world of electronics industries low II. LITERATURE REVIEW power has emerged as a principal theme. Power dissipation has become an essential consideration as area One of the promising computing technologies assuring for VLSI Chip design and performance. With shrinking zero power dissipation is the Reversible Logic. Reversible technology reducing overall power management and Logic has a wide spectrum of applications like power consumption on chip are the main challenges Nanotechnology based systems, Bio Informatics optical below 100nm as a result of increased complexity. When Circuits, quantum computing, and Low Power VLSI. it comes to many designs available, power optimization is Therefore, this paper represents the 4 bit reversible essential as timing as a result to the demand to extend comparator based on classical logic circuit which uses the battery life and reduce package cost. Asto reduce the existing reversible gates. The design simply aims at power consumption we have many techniques from those reducing optimization parameters like quantum cost, techniques, lets we implement on Reverse Logic. In garbage outputs, and the number of constant inputs. The reversible logic it as greater advantage and executing in results obtained in this research work indicate that the high speed, with less power dissipation which is proposed comparator has one constant input and 4 quantum implemented by using basic logic gates in less area.
    [Show full text]
  • Gray Codes and Efficient Exhaustive Generation for Several Classes Of
    Gray codes and efficient exhaustive generation for several classes of restricted words Ahmad Sabri To cite this version: Ahmad Sabri. Gray codes and efficient exhaustive generation for several classes of restricted words. Other [cs.OH]. Université de Bourgogne, 2015. English. NNT : 2015DIJOS007. tel-01203339 HAL Id: tel-01203339 https://tel.archives-ouvertes.fr/tel-01203339 Submitted on 22 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thèse de Doctorat école doctorale sciences pour l’ingénieur et microtechniques UNIVERSITÉ DE BOURGOGNE Gray codes and efficient exhaustive generation for several classes of restricted words n AHMAD SABRI Thèse de Doctorat école doctorale sciences pour l’ingénieur et microtechniques UNIVERSITÉ DE BOURGOGNE THESE` present´ ee´ par AHMAD SABRI pour obtenir le Grade de Docteur de l’Universite´ de Bourgogne Specialit´ e´ : Informatique Gray codes and efficient exhaustive generation for several classes of restricted words Unite´ de Recherche : Laboratoire Electronique, Informatique et Image (LE2I) Soutenue publiquement
    [Show full text]
  • A Gray Code for Combinations of a Multiset
    AGray Co de for Combinations of a Multiset y Frank Ruskey Carla Savage Department of Computer Science Department of Computer Science University of Victoria North Carolina State University Victoria BC VW P Canada Raleigh North Carolina Abstract Let Ck n n n denote the set of all k elementcombinations of the t multiset consisting of n o ccurrences of i for i tEach combination i is itself a multiset For example C ff g f g f g f gg Weshowthatmultiset combinations can b e listed so that successive combina ultaneously generalize tions dier by one element Multiset combinations sim for which minimal change listings called Gray co des are known comp ositions and combinations Research supp orted by NSERC grantA y Research supp orted by National Science Foundation GrantNo CCR and National Security Agency Grant No MDAH Intro duction By Ck n n n we denote the set of all ordered t tuples x x x t t satisfying x x x k where x n for i t t i i We assume throughout the pap er that n is a p ositiveinteger for i t i Solutions to are referred to as combinations of a multiset b ecause they can b e regarded as k subsets of the multiset consisting of n copies of i for i t i Solution x x x corresp onds to the k subset consisting of x copies of i for t i i t For example C corresp onds to ff g f g f g f gg When each n is the set Ck n n n is simply the set of all combinations of i t t elements chosen k at a time Alternatively the elements of Ck n n n can b e regarded as placements t of k identical balls into
    [Show full text]
  • Gray Codes and Puzzles on Iterated Complete Graphs
    GRAY CODES AND PUZZLES ON ITERATED COMPLETE GRAPHS ELIZABETH WEAVER ADVISOR: PAUL CULL OREGON STATE UNIVERSITY ABSTRACT. There exists an infinite family of puzzles, the SF Puzzle, that correspond to labelings on iterated complete graphs of odd dimensions, the simplest being the well-known Towers of Hanoi puzzle. There is also a puzzle, knownas Spin-Out that correspondsto the reflected binary Gray code; however, thus far there has been no family of puzzles defined that can be represented by labelings on all iterated complete graphs of even dimensions. This paper explores the properties that labelings of known puzzles possess, tries to determine the necessary labeling properties for a new puzzle, and n defines a puzzle on K4 . Unfortunately, this puzzle does not extend to labelings on all even dimension iterated complete graphs. 1. INTRODUCTION Mathematics forms the basis for many well-known and interesting puzzles. For example, the popular game the Towers of Hanoi can be represented graphically and has several applications in graph theory and in error correcting codes [2]. By mathematically generalizing the rules of the Towers of Hanoi, a new family of puzzles was created that can be easily represented by graphs [5]. In these puzzles there are an odd number of possible positions for each piece and an easily defined rule to determine the possible moves for each piece. It seems logical that a family of puzzles should exist where there are an even number d possible moves for each piece, but with the exception of d = 2 no one rule seems to describe a puzzle like that of the Towers of Hanoi.
    [Show full text]