DOCSLIB.ORG

Nigel Boston Claude Shannon Institute University College Dublin Extended Binary Golay Code

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nigel Boston Claude Shannon Institute University College Dublin Extended Binary Golay Code Golay Pseudocodewords Nigel Boston Claude Shannon Institute University College Dublin Extended Binary Golay Code • Encodes 12 bits of data in a 24-bit word so that any 3-bit error is corrected and any 4- bit error detected • Particular 12-diml subspace of GF(2)24 • Automorphism group is Mathieu group M24 Voyager 1 and 2 (1979 and 1980) • Transmitted hundreds of colour pictures of Jupiter and Saturn • US standards for ALE (automatic link establishment) in high frequency systems use this code for forward error correction Generator Matrix Properties of the Code • It is the span of the 12 rows of that matrix • Every vector ≠ 0 has ≥ 8 ones (weight ≥ 8) • It is therefore a [24,12,8] binary linear code • Its weight enumerator (which counts words of each weight) is: x24 + 759x16t8 + 2576x12t12 + 759x8t16 + t24 Gleason’s Theory (ICM 1970) • If C is a self-dual binary linear code, then by MacWilliams its weight enumerator W is invariant under (x,t) a 2-1/2(x+t,x-t) • If all weights are divisible by 4, then W is also invariant under (x,t) a (x,jt) • W is invariant under a group of order 192 • Follows W is a polynomial in the weight enumerators of Hamming and Golay codes Broue-Enguehard Isomorphism • Algebra isomorphism: modular forms of level one, weight divisible by 4 to ring containing weight enumerators • Has been generalized in many directions, e.g Siegel modular forms, half-integral weight,… Tail-Biting Trellises Theory of TBT’s • That’s a TBT for the [8,4,4] Hamming code • The labels on paths that start and finish at the same point yield the codewords • The TBT approach allows for important decoding algorithms (forward-backward, Viterbi, …) • Pseudocodewords give suboptimal decoding Pseudocodewords • Unwrapping a TBT G gives a conventional trellis, called a computation tree G’ • The min-sum algorithm on G’ behaves as the iterative algorithm on G • Pseudocodewords satisfy constraints of G’ • Codeword = pseudocodeword of period 1 • There exist pseudocodewords of period >1 Pseudoweight • Wiberg found a formula for the effective weight of a pseudocodeword for the AWGN 2 2 channel, namely (∑ nj) /∑ nj • Here nj is the number of edges of G’ above the jth edge of G that have label one • Example: 01 01 01 10 followed by 11 00 10 10 labels a pseudocodeword of period 2, namely 12 01 11 20 - pseudowt 82/12 = 5.33 Minimal TBT for Golay Code • In 1999, Calderbank, Forney, and Vardy solved an old problem asking for a 16-state TBT for the extended binary Golay code • Join together three copies of the TBT on the left Minimal Pseudoweight? • Codeword of weight 8 has pseudoweight 8 • Fundamental question- is there a nonzero pseudocodeword of pseudoweight <8 ? • Subject of private emails among experts • Example (period 2): nj=1 in 8 places,=2 in 2 places, =0 in 14 places- pseudowt 122/16=9 • Example (period 3): nj=1 in 6 places,=2 in 2 places,=3 in 2 places, else =0. Get 162/32=8 Period 2 Pseudocodewords • The labels on the paths of period 2 form a [48,24,8] binary linear code C2 • The generator matrix given earlier for the Golay code is a TOGM (trellis-oriented generator matrix) • We can use this to obtain a TOGM for C2 given on the next slide Generator Matrix for C2 Pseudoweight Distribution • 224 is small enough for all pseudoweights to be computed • No pseudocodeword of period 2 and pseudoweight < 8 • For period n, have a [24n,12n,8] binary linear code Cn too big! Words of Low Weight • The 1053 words of C2 of pseudoweight 8 are the 759 Golay words of weight 8 plus the 294 words of C2 of weight 8 • Cn has 147n words of weight 8 (n > 1) • The weight enumerator of C2 is: x48 + 294x40t8 +14944x36t12 + 543327x32t16 + 3978912x28t20 + 7702260x24t24 + 3978912x20t28 + 543327x16t32 + 14944x12t36 + 294x8t40 + t48 Conjecture • I conjecture that the set of pseudoweights that arise from codewords in Cn of weight w is independent of n. Call this set S(w) • Conjecture: S(8)= {8.00}, S(12)={9.00,12.00}, S(16)={8.00, 9.14,10.67,12.80,16.00}, S(20)={9.09,10.00,11.11,12.50,14.29,16.67}, S(24)={8.00,9.00,9.60,10.29,11.08,12.00,13.09, 14.40,16.00,18.00}, … Better Weight Enumerator • The pseudocodeword 12 02 10 11 01 21 12 20 10 00 00 02 has 10 zeros, 8 ones, 6 twos • Attach to this the monomial x6 y8 t10 • Sum over all 224 pseudocodewords to get a multivariate weight enumerator W2(x,y,t) where W (x,0,t) is the weight enumerator of 2 2 the Golay code and x=y gives that of C2 Multivariate Weight Enumerator 24 16 8 16 8 14 8 2 W2(x,y,t) = x + 294x y + 759x t + 9792x y t + 5152x12y12 + 178248x12y8t4 + 2576x12t12 + 340032x10y12t2 + 748608x10y8t6 + 24288x8y16 + 2550240x8y12t4 + 1234980x8y8t8 + 759x8t16 + 680064x6y16t2 + 4760448x6y12t6 + 748608x6y8t10 + 1700160x4y16t4 + 2550240x4y12t8 + 178248x4y8t12 + 680064x2y16t6 + 340032x2y12t10 + 9792x2y8t14 + 4096y24 + 24288y16t8 + 5152y12t12 + 294y8t16 + t24 Properties of New Enumerator • In addition to those noted already, W2(x,y,t) is symmetric under x ↔ t. By observing which monomials arise, we can tell that no pseudocodeword has pseudoweight < 8 8 8 • W2(x,y,t) = G + 294 y r(16) - 25488 y (r(1)r(2)r(4))2 + 5152 y12 r(12) + 24288y16 r(8), where G is the Golay enumerator and r(n) the even weight codes’ ((x+t)n+(x-t)n)/2 Nicer Formula Let J(x,y,t) = x24 + y24 + t24 + 759(x16y8+x8y16+x16t8+x8t16+y8t16+y16t8) + 12 12 12 12 12 12 8 8 8 2576(x y +x t +y t ) + 3186x y t 12 Then W2(x,y,t) = 2 J((x+t)/2, y, (x-t)/2) 2 The curve in P cut out by W2(x,y,t) = 0 has large automorphism group, order 384 Period 3 Pseudocodewords • Define W3(x,y,z,t) similarly • Computing low weight codewords gives: 24 16 8 16 8 W3(x,y,z,t) = x + 441x y + 513x z + 759x16t8 + 7560x14y8z2 + 288x14y6z2t2 + … 36 There are too many pseudocodewords (2 ) to allow computation of W3 by brute force Symmetry under x ↔ t, y ↔ z New Invariant Theory n n-1 n-j j n • Let x0=x , x1=x t, …, xj=x t , …, xn=t • Gleason’s maps (x,t) a 2-1/2(x+t,x-t) and (x,t) a (x,jt) induce maps on (x0,…,xn) • For n=2, these plus (x,y,t) a (x,y,-t) generate 384 maps leaving W2 invariant • For n=3, these plus (x,y,z,t) a (x,y,-z,-t) generate 1152 maps leaving W3 invariant New Invariant Theory II • For each n, get a group Gn of (n+1) by (n+1) complex matrices that leave Wn invariant • MAGMA computes the homogeneous polynomials of degree 24 in n+1 variables invariant under Gn • Wn is a simple linear combination of these Weight Enumerator W3 24 16 8 16 8 16 8 14 8 2 W3=x +441x y +513x z +759x t +7560x y z + 288x14y6z2t2+14112x14z10+11520x13y7z3t+2304x13 y5z3t3+4608x12y12+792x12y10t2+132192x12y8z4+14 4x12y8t4+239040x12y6z4t2+132840x12y4z4t4+17085 6x12z12+2576x12t12+9504x11y11zt+23040x11y9zt3+1 456704x11y7z5t+2304x11y7zt5+1456704x11y5z5t3+1 8432x11y3z5t5+237384x10y12z2+676080x10y10z2t2+ 3406968x10y8z6+700920x10y8z2t4+11168928x10y6z 6t2+213120x10y6z2t6+3016080x10y4z6t4+213984x10 y2z6t6+1118880x10z14+3238272x9y11z3t+7850016x 9y9z3t3 +39790656x9y7z7t+3196800x9y7z3t5 + … W3 continued … +49955130y8z8t8+513y8t16+79634016y6z16t2 +109442880y6z12t6+3406968y6z8t10+220676 40y4z16t4+6877440y4z12t8+132192y4z8t12+12 52800y2z16t6+237384y2z12t10+7560y2z8t14+3 3291z24+16371z16t8+4608z12t12+441z8t16+t24 212 explicit terms, all with pseudoweight ≥ 8 Period 4 Pseudocodewords • G4 ≤ GL(5,C) has order 384 • MAGMA finds a basis of 153 invariant polynomials of degree 24 • 87 have every monomial of pseudowt ≥ 8 • Exclude others using unique monomials coming from words of C4 of weight ≤ 24 • 248 pseudocodewords all have pseudowt ≥ 8 Remarks • There exist codes with AWGN pseudoweights less than the minimum distance • There are pseudoweights for other channels - for the BSC, it’s twice the median of the descendingly sorted nj. There exist Golay pseudocodewords with BSC pseudowt < 8.
Load more

Recommended publications
  • Mceliece Cryptosystem Based on Extended Golay Code
    McEliece Cryptosystem Based On Extended Golay Code Amandeep Singh Bhatia∗ and Ajay Kumar Department of Computer Science, Thapar university, India E-mail: ∗[email protected] (Dated: November 16, 2018) With increasing advancements in technology, it is expected that the emergence of a quantum computer will potentially break many of the public-key cryptosystems currently in use. It will negotiate the confidentiality and integrity of communications. In this regard, we have privacy protectors (i.e. Post-Quantum Cryptography), which resists attacks by quantum computers, deals with cryptosystems that run on conventional computers and are secure against attacks by quantum computers. The practice of code-based cryptography is a trade-off between security and efficiency. In this chapter, we have explored The most successful McEliece cryptosystem, based on extended Golay code [24, 12, 8]. We have examined the implications of using an extended Golay code in place of usual Goppa code in McEliece cryptosystem. Further, we have implemented a McEliece cryptosystem based on extended Golay code using MATLAB. The extended Golay code has lots of practical applications. The main advantage of using extended Golay code is that it has codeword of length 24, a minimum Hamming distance of 8 allows us to detect 7-bit errors while correcting for 3 or fewer errors simultaneously and can be transmitted at high data rate. I. INTRODUCTION Over the last three decades, public key cryptosystems (Diffie-Hellman key exchange, the RSA cryptosystem, digital signature algorithm (DSA), and Elliptic curve cryptosystems) has become a crucial component of cyber security. In this regard, security depends on the difficulty of a definite number of theoretic problems (integer factorization or the discrete log problem).
    [Show full text]
  • The Theory of Error-Correcting Codes the Ternary Golay Codes and Related Structures
    Math 28: The Theory of Error-Correcting Codes The ternary Golay codes and related structures We have seen already that if C is an extremal [12; 6; 6] Type III code then C attains equality in the LP bounds (and conversely the LP bounds show that any ternary (12; 36; 6) code is a translate of such a C); ear- lier we observed that removing any coordinate from C yields a perfect 2-error-correcting [11; 6; 5] ternary code. Since C is extremal, we also know that its Hamming weight enumerator is uniquely determined, and compute 3 3 3 3 3 3 12 6 6 3 9 12 WC (X; Y ) = (X(X + 8Y )) − 24(Y (X − Y )) = X + 264X Y + 440X Y + 24Y : Here are some further remarkable properties that quickly follow: A (5,6,12) Steiner system. The minimal words form 132 pairs fc; −cg with the same support. Let S be the family of these 132 supports. We claim S is a (5; 6; 12) Steiner system; that is, that any pentad (5-element 12 6 set of coordinates) is a subset of a unique hexad in S. Because S has the right size 5 5 , it is enough to show that no two hexads intersect in exactly 5 points. If they did, then we would have some c; c0 2 C, both of weight 6, whose supports have exactly 5 points in common. But then c0 would agree with either c or −c on at least 3 of these points (and thus on exactly 4, because (c; c0) = 0), making c0 ∓ c a nonzero codeword of weight less than 5 (and thus exactly 3), which is a contradiction.
    [Show full text]
  • Review of Binary Codes for Error Detection and Correction
    ` ISSN(Online): 2319-8753 ISSN (Print): 2347-6710 International Journal of Innovative Research in Science, Engineering and Technology (A High Impact Factor, Monthly, Peer Reviewed Journal) Visit: www.ijirset.com Vol. 7, Issue 4, April 2018 Review of Binary Codes for Error Detection and Correction Eisha Khan, Nishi Pandey and Neelesh Gupta M. Tech. Scholar, VLSI Design and Embedded System, TCST, Bhopal, India Assistant Professor, VLSI Design and Embedded System, TCST, Bhopal, India Head of Dept., VLSI Design and Embedded System, TCST, Bhopal, India ABSTRACT: Golay Code is a type of Error Correction code discovered and performance very close to Shanon‟s limit . Good error correcting performance enables reliable communication. Since its discovery by Marcel J.E. Golay there is more research going on for its efficient construction and implementation. The binary Golay code (G23) is represented as (23, 12, 7), while the extended binary Golay code (G24) is as (24, 12, 8). High speed with low-latency architecture has been designed and implemented in Virtex-4 FPGA for Golay encoder without incorporating linear feedback shift register. This brief also presents an optimized and low-complexity decoding architecture for extended binary Golay code (24, 12, 8) based on an incomplete maximum likelihood decoding scheme. KEYWORDS: Golay Code, Decoder, Encoder, Field Programmable Gate Array (FPGA) I. INTRODUCTION Communication system transmits data from source to transmitter through a channel or medium such as wired or wireless. The reliability of received data depends on the channel medium and external noise and this noise creates interference to the signal and introduces errors in transmitted data.
    [Show full text]
  • A Parallel Algorithm for Query Adaptive, Locality Sensitive Hash
    A CUDA Based Parallel Decoding Algorithm for the Leech Lattice Locality Sensitive Hash Family A thesis submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in the School of Electric and Computing Systems of the College of Engineering and Applied Sciences November 03, 2011 by Lee A Carraher BSCE, University of Cincinnati, 2008 Thesis Advisor and Committee Chair: Dr. Fred Annexstein Abstract Nearest neighbor search is a fundamental requirement of many machine learning algorithms and is essential to fuzzy information retrieval. The utility of efficient database search and construction has broad utility in a variety of computing fields. Applications such as coding theory and compression for electronic commu- nication systems as well as use in artificial intelligence for pattern and object recognition. In this thesis, a particular subset of nearest neighbors is consider, referred to as c-approximate k-nearest neighbors search. This particular variation relaxes the constraints of exact nearest neighbors by introducing a probability of finding the correct nearest neighbor c, which offers considerable advantages to the computational complexity of the search algorithm and the database overhead requirements. Furthermore, it extends the original nearest neighbors algorithm by returning a set of k candidate nearest neighbors, from which expert or exact distance calculations can be considered. Furthermore thesis extends the implementation of c-approximate k-nearest neighbors search so that it is able to utilize the burgeoning GPGPU computing field. The specific form of c-approximate k-nearest neighbors search implemented is based on the locality sensitive hash search from the E2LSH package of Indyk and Andoni [1].
    [Show full text]
  • On the Algebraic Structure of Quasi Group Codes
    ON THE ALGEBRAIC STRUCTURE OF QUASI GROUP CODES MARTINO BORELLO AND WOLFGANG WILLEMS Abstract. In this note, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi group codes, that is as linear codes with a free group of permutation automorphisms. An algebraic description, including the concatenated structure, of such codes is presented. Finally, self-duality of quasi group codes is investigated. 1. Introduction In the theory of error correcting codes, the linear ones play a central role for their algebraic properties which allow, for example, their easy description and storage. Quite early in their study, it appeared convenient to add additional algebraic structure in order to get more information about the parameters and to facilitate the decoding process. In 1957, E. Prange introduced the now well- known class of cyclic codes [18], which are the forefathers of many other families of codes with symmetries introduced later. In particular, abelian codes [2], group codes [15], quasi-cyclic codes [9] and quasi-abelian codes [19] are distinguished descendants of cyclic codes: they have a nice algebraic structure and many optimal codes belong to these families. The aim of the current note is to give an intrinsic description of these classes in terms of their permutation automorphism groups: we will show that all of them have a free group of permutation automorphisms (which is not necessarily the full permutation automorphism group of the code). We will define a code with this property as a quasi group code. So all families cited above can be described in this way.
    [Show full text]
  • Error Detection and Correction Using the BCH Code 1
    Error Detection and Correction Using the BCH Code 1 Error Detection and Correction Using the BCH Code Hank Wallace Copyright (C) 2001 Hank Wallace The Information Environment The information revolution is in full swing, having matured over the last thirty or so years. It is estimated that there are hundreds of millions to several billion web pages on computers connected to the Internet, depending on whom you ask and the time of day. As of this writing, the google.com search engine listed 1,346,966,000 web pages in its database. Add to that email, FTP transactions, virtual private networks and other traffic, and the data volume swells to many terabits per day. We have all learned new standard multiplier prefixes as a result of this explosion. These days, the prefix “mega,” or million, when used in advertising (“Mega Cola”) actually makes the product seem trite. With all these terabits per second flying around the world on copper cable, optical fiber, and microwave links, the question arises: How reliable are these networks? If we lose only 0.001% of the data on a one gigabit per second network link, that amounts to ten thousand bits per second! A typical newspaper story contains 1,000 words, or about 42,000 bits of information. Imagine losing a newspaper story on the wires every four seconds, or 900 stories per hour. Even a small error rate becomes impractical as data rates increase to stratospheric levels as they have over the last decade. Given the fact that all networks corrupt the data sent through them to some extent, is there something that we can do to ensure good data gets through poor networks intact? Enter Claude Shannon In 1948, Dr.
    [Show full text]
  • Error-Correction and the Binary Golay Code
    London Mathematical Society Impact150 Stories 1 (2016) 51{58 C 2016 Author(s) doi:10.1112/i150lms/t.0003 e Error-correction and the binary Golay code R.T.Curtis Abstract Linear algebra and, in particular, the theory of vector spaces over finite fields has provided a rich source of error-correcting codes which enable the receiver of a corrupted message to deduce what was originally sent. Although more efficient codes have been devised in recent years, the 12- dimensional binary Golay code remains one of the most mathematically fascinating such codes: not only has it been used to send messages back to earth from the Voyager space probes, but it is also involved in many of the most extraordinary and beautiful algebraic structures. This article describes how the code can be obtained in two elementary ways, and explains how it can be used to correct errors which arise during transmission 1. Introduction Whenever a message is sent - along a telephone wire, over the internet, or simply by shouting across a crowded room - there is always the possibly that the message will be damaged during transmission. This may be caused by electronic noise or static, by electromagnetic radiation, or simply by the hubbub of people in the room chattering to one another. An error-correcting code is a means by which the original message can be recovered by the recipient even though it has been corrupted en route. Most commonly a message is sent as a sequence of 0s and 1s. Usually when a 0 is sent a 0 is received, and when a 1 is sent a 1 is received; however occasionally, and hopefully rarely, a 0 is sent but a 1 is received, or vice versa.
    [Show full text]
  • The Binary Golay Code and the Leech Lattice
    The binary Golay code and the Leech lattice Recall from previous talks: Def 1: (linear code) A code C over a field F is called linear if the code contains any linear combinations of its codewords A k-dimensional linear code of length n with minimal Hamming distance d is said to be an [n, k, d]-code. Why are linear codes interesting? ● Error-correcting codes have a wide range of applications in telecommunication. ● A field where transmissions are particularly important is space probes, due to a combination of a harsh environment and cost restrictions. ● Linear codes were used for space-probes because they allowed for just-in-time encoding, as memory was error-prone and heavy. Space-probe example The Hamming weight enumerator Def 2: (weight of a codeword) The weight w(u) of a codeword u is the number of its nonzero coordinates. Def 3: (Hamming weight enumerator) The Hamming weight enumerator of C is the polynomial: n n−i i W C (X ,Y )=∑ Ai X Y i=0 where Ai is the number of codeword of weight i. Example (Example 2.1, [8]) For the binary Hamming code of length 7 the weight enumerator is given by: 7 4 3 3 4 7 W H (X ,Y )= X +7 X Y +7 X Y +Y Dual and doubly even codes Def 4: (dual code) For a code C we define the dual code C˚ to be the linear code of codewords orthogonal to all of C. Def 5: (doubly even code) A binary code C is called doubly even if the weights of all its codewords are divisible by 4.
    [Show full text]
  • Linear Codes and Syndrome Decoding
    Linear Codes and Syndrome Decoding José Luis Gómez Pardo Departamento de Matemáticas, Universidade de Santiago de Compostela 15782 Santiago de Compostela, Spain e-mail: [email protected] Introduction This worksheet contains an implementation of the encoding and decoding algorithms associated to an error-correcting linear code. Such a code can be characterized by a generator matrix or by a parity- check matrix and we give procedures to compute a generator matrix from a spanning set of the code and to compute a parity-check matrix from a generator matrix. We introduce, as examples, the [7, 4, 2] binary Hamming code, the [24, 12, 8] and [23, 12, 7] binary Golay codes and the [12, 6, 6] and [11, 6, 5] ternary Golay codes. We give procedures to compute the minimum distance of a linear code and we use them with the Hamming and Golay codes. We show how to build the standard array and the syndrome array of a linear code and we give an implementation of syndrome decoding. In the last section, we simulate a noisy channel and use the Hamming and Golay codes to show how syndrome decoding allows error correction on text messages. The references [Hill], [Huffman-Pless], [Ling], [Roman] and [Trappe-Washington] provide detailed information about linear codes and, in particular, about syndrome decoding. Requirements and Copyright Requirements: Maple 16 or higher © (2019) José Luis Gómez Pardo Generator matrices and parity-check matrices Linear codes are vector spaces over finite fields. Maple has a package called LinearAlgebra:- Modular (a subpackage of the LinearAlgebra m (the integers modulo a positive integer m, henceforth denoted , whose elements we will denote by {0, 1, ..
    [Show full text]
  • Iterative Decoding of Product Codes
    Iterative Decoding of Product Codes OMAR AL-ASKARY RADIO COMMUNICATION SYSTEMS LABORATORY Iterative Decoding of Product Codes OMAR AL-ASKARY A dissertation submitted to the Royal Institute of Technology in partial fulfillment of the requirements for the degree of Licentiate of Technology April 2003 TRITA—S3—RST—0305 ISSN 1400—9137 ISRN KTH/RST/R--03/05--SE RADIO COMMUNICATION SYSTEMS LABORATORY DEPARTMENT OF SIGNALS, SENSORS AND SYSTEMS Abstract Iterative decoding of block codes is a rather old subject that regained much inter- est recently. The main idea behind iterative decoding is to break up the decoding problem into a sequence of stages, iterations, such that each stage utilizes the out- put from the previous stages to formulate its own result. In order for the iterative decoding algorithms to be practically feasible, the complexity in each stage, in terms of number of operations and hardware complexity, should be much less than that for the original non-iterative decoding problem. At the same time, the perfor- mance should approach the optimum, maximum likelihood decoding performance in terms of bit error rate. In this thesis, we study the problem of iterative decoding of product codes. We propose an iterative decoding algorithm that best suits product codes but can be applied to other block codes of similar construction. The algorithm approaches maximum likelihood performance. We also present another algorithm which is suboptimal and can be viewed as a practical implementation of the first algorithm on product codes. The performance of the suboptimal algorithm is investigated both analytically and by computer simulations. The complexity is also investigated and compared to the complexity of GMD and Viterbi decoding of product codes.
    [Show full text]
  • Constructing Good Covering Codes for Applications in Steganography
    Constructing good covering codes for applications in Steganography J¨urgen Bierbrauer1 and Jessica Fridrich2 1 Department of Mathematical Sciences Michigan Technological University HOUGHTON (MI) 49931 (USA) 2 Department of Electrical and Computer Engineering Binghamton University BINGHAMTON (NY) 13902-6000 Abstract. Application of covering codes to data embedding improves embedding efficiency and security of steganographic schemes. In this pa- per, we describe several familes of covering codes constructed using the blockwise direct sum of factorizations. We show that non-linear construc- tions offer better performance compared to simple linear covering codes currently used by steganographers. Implementation details are given for a selected code family. 1 Introduction Steganography is the art of stealth communication. Its purpose is to make com- munication undetectable. The steganography problem is also known as the pris- oners’ dilemma formulated by Simmons [28]. Alice and Bob are imprisoned and want to hatch an escape plan. They are allowed to communicate via a channel monitored by a warden. If the warden finds out that they are communicating secretly, he throws them into solitary confinement. Thus, the prisoners need to design a method to exchange messages without raising the warden’s suspicion. The prisoners hide their messages in innocuous-looking cover objects by slightly modifying them (obtaining stego objects). The embedding process is usually driven by a stego key, which is a secret shared between Alice and Bob. It is typically used to select a subset of the cover object and the order in which the cover object elements are visited during embedding. The most important property of any steganographic communication is sta- tistical undetectability.
    [Show full text]
  • Insertion/Deletion Error Correction Using Path Pruned Convolutional Codes and Extended Prefix Codes
    Insertion/Deletion Error Correction using Path Pruned Convolutional Codes and Extended Prefix Codes By Muhammad Waqar Saeed A research report submitted to the faculty of Engineering and Build Environment In partial fulfilment of the requirements for the degree Masters of Science in Engineering in Electrical Engineering (Telecommunication) at the UNIVERSITY OF THE WITWATERSRAND, JOHANNESBURG Supervisor: Dr. Ling Cheng August 2014 i Declaration I declare that this project report is my own, unaided work, except where otherwise acknowledged. It is being submitted for the partial fulfilment of the degree of Master of Science in Engineering at the University of the Witwatersrand, Johannesburg, South Africa. It has not been submitted before for any degree or examination at any other university. Candidate Signature : ................................................................ Name : ................................................................. Date : (Day).......... (Month).......... (Year)............ ii Abstract Synchronization error correction has been under discussion since the early development of coding theory. In this research report a novel coding system based on the previous done work on path- pruned convolutional codes and extended prefix synchronization codes is presented. This new coding scheme is capable of correcting insertion, deletion and synchronization errors. A codebook has been constructed that contains synchronization patterns made up of a constraint part (maker sequence) and an unconstraint part based on the concept of extended prefix codes. One of these synchronization error patterns are padded in front of each frame. This process is done by mapping information bit to a corresponding bit sequence using a mapping table. The mapping table is constructed by using path-pruning process. An original rate convolutional code is first punctured using a desired puncturing matrix to make enough paths available at each state of the trellis.
    [Show full text]