Conway and Sloane (1990) have listed the possible weight enumerators for extremal self-dual codes... more Conway and Sloane (1990) have listed the possible weight enumerators for extremal self-dual codes up to length 72. In this correspondence, we construct extremal singly-even self-dual [60,30,12] codes whose weight enumerator does not appear in this list. In addition, we present the possible weight enumerators for extremal self-dual codes of length 60
A number of misconceptions and ill-suited approaches concerning fast frequency-hopped (FH) M-ary ... more A number of misconceptions and ill-suited approaches concerning fast frequency-hopped (FH) M-ary NCFSK have risen that have led to the concealment of the impressive performance enhancement capability of some diversity combining methods in the presence of strong intelligent jamming, including partial-band noise (PBN) and multiple tone (MT). Contrary to the conventional wisdom, practical fast FH for anti-jam (AJ) applications uses a constant hop rate whereas the data rate is backed off as needed to provide diversity capability. Then, there is no optimum diversity and the energy per hop is a better measure than the traditional energy per bit used in nonjammed systems. It is then possible to operate under much higher jamming levels than with just the basic FH and standard error-correction techniques alone. These diversity combining techniques are relatively easy to implement, and have been implemented, in real time. Processing gain is an inappropriate measure of performance when applied to non-Gaussian jamming. The perception of a noncoherent combining loss for diversity combining is shown not only to be false, but, for the types of jamming considered, diversity combining actually performs better than coherent combining. Side information is shown to be of little or no value for the types of jamming considered. For large jamming levels, PBN jamming generally causes more degradation in performance than MT jamming
up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search... more up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search or linear programming, or are those which meet the upper bound in [8]. These results are further evidence that the class of QC codes is good, especially in light of the fact that we have only considered 1-generator QC codes.
The class of Quasi-Cyclic Error Correcting Codes is investigated. It is shown that they contain m... more The class of Quasi-Cyclic Error Correcting Codes is investigated. It is shown that they contain many of the best known binary and nonbinary codes. Tables of rate 1/p and (p − 1)/p Quasi-Cyclic (QC) codes are constructed, which are a compilation of previously best known codes as well as many new codes constructed using exhaustive, and other more sophisticated search techniques. Many of these binary codes attain the known bounds on the maximum possible minimum distance, and 13 improve the bounds. The minimum distances and generator polynomials of all known best codes are given. The search methods are outlined and the weight divisibility of the codes is noted. The weight distributions of some s-th Power Residue (PR) codes and related rate 1/s QC codes are found using the link established between PR codes and QC codes. Subcodes of the PR codes are found by deleting certain circulant matrices in the corresponding QC code. They are used as a starting set of circulants for other techniques. Nonbinary Power Residue codes and related QC codes are constructed over GF(3), GF(4), GF(5), GF(7) and GF(8). Their subcodes are also used to find good nonbinary QC codes. A simple and efficient algorithm for constructing primitive polynomials with linearly independent roots over the Galois Field of q elements, GF(q), is developed. Tables of these polynomials are presented. These Tables are unknown for polynomials with nonbinary coefficients, and the known binary Tables are incomplete. The polynomials are employed in such diverse areas as construction of error correcting codes, efficient VLSI implementation of multiplication and inverse operations over Galois Fields, and digital testing of integrated circuits. Using the link established between generalized tail biting convolutional codes and binary QC codes, good QC codes are constructed based on iii Optimum Distance Profile (ODP) convolutional codes. Several best rate 2/3 systematic codes up to circulant size 20 are constructed in this manner.
IEEE Journal on Selected Areas in Communications, 1993
research interests focus on microstrip patch antennas and arrays, monolithic mircrowave integrate... more research interests focus on microstrip patch antennas and arrays, monolithic mircrowave integrated circuits, and code synchronizers for radar and spreadspectrum systems. He has acted for years as a consultant to CAL Corporation, and has conducted many joint research programs with the Communications Research Centre and the Defence Research Establishment Ottawa over the past 16 years.
Let [n; k; d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d ov... more Let [n; k; d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF (q). The following quasicyclic codes are constructed in this paper:
Formally self-dual even codes have recently been studied. Double circulant even codes are a famil... more Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes.
up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search... more up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search or linear programming, or are those which meet the upper bound in [8]. These results are further evidence that the class of QC codes is good, especially in light of the fact that we have only considered 1-generator QC codes.
Given a quasi-cyclic code with minimum Hamming distance d, a set of convolutional codes is derive... more Given a quasi-cyclic code with minimum Hamming distance d, a set of convolutional codes is derived with free distance equal to d. It is shown that an increase in the rate of these codes results in a decrease in the memory length. The connection between these codes is illustrated with several well-known quasi-cyclic codes. The free distance of some partial unit memory convolutional codes can be determined using the results in this correspondence.
A mobile ad hoc network (MANET) is a collection of mobile nodes that form a wireless network with... more A mobile ad hoc network (MANET) is a collection of mobile nodes that form a wireless network without the use of a fixed infrastructure or centralized administration, and every node acts as a host as well as a router. The topology of an ad hoc network changes frequently and unpredictably. The mobile nature and dynamic topology of MANETs make it very difficult to provide Quality-of-Service (QoS) assurance in such networks. Considering the limited bandwidth and battery power, finding routes that satisfy the bandwidth constraint of applications is a significant challenge.
In this letter, we propose a new modulation scheme called pulse position amplitude modulation (PP... more In this letter, we propose a new modulation scheme called pulse position amplitude modulation (PPAM) for ultra-wideband (UWB) communication systems. PPAM combines pulse position modulation and pulse amplitude modulation to provide good system performance and low computational complexity. The channel capacity of PPAM is determined for a time-hopping multiple-access UWB communication system. The error probability and performance bounds are derived
IEEE Transactions on Wireless Communications, 2005
The capacity and error probability of orthogonal space-time block codes (STBCs) are considered fo... more The capacity and error probability of orthogonal space-time block codes (STBCs) are considered for pulse-amplitude modulation/phase shift keying/quadrature-amplitude modulation (PAM/PSK/QAM) in fading channels. The approach is based on an equivalent scalar additive white Gaussian noise channel with a channel gain proportional to the Frobenius norm of the matrix channel for the STBC. Using this effective channel, capacity and probability of error expressions are derived for PSK/PAM/QAM modulation with space-time block coding. Rayleigh-, Ricean-, and Nakagami-fading channels are considered. As an application, these results are extended to obtain the capacity and probability of error for a multiuser direct sequence code-division multiple-access system employing space-time block coding.
The class of quasi-twisted (QT) codes is a generalization of the class of quasi-cyclic codes, sim... more The class of quasi-twisted (QT) codes is a generalization of the class of quasi-cyclic codes, similar to the way constacyclic codes are a generalization of cyclic codes. In this paper, rate 1/p QT codes over GF(3) are presented which have been constructed using integer linear programming and heuristic combinatorial optimization. Many of these attain the maximum possible minimum distance for any linear code with the given parameters, and several improve the maximum known minimum distances. Two of these new codes, namely (90, 6, 57) and (120, 6, 78), are optimal and so prove that d3(90, 6)=57 and d3(120, 6)=78
Previous results have shown that the class of quasi-cyclic (QC) codes contains many good codes. I... more Previous results have shown that the class of quasi-cyclic (QC) codes contains many good codes. In this paper, new rate (m-1)/pm QC codes over GF(3) and GF(4) are presented. These codes have been constructed using integer linear programming and a heuristic combinatorial optimization algorithm based on a greedy local search. Most of these codes attain the maximum possible minimum distance for any linear code with the same parameters, i.e., they are optimal, and 58 improve the maximum known distances. The generator polynomials for these 58 codes are tabulated, and the minimum distances of rate (m-1)/pm QC codes are given.
nternet Protocol TV (IPTV) is considered to be the next killer application. With two-way communic... more nternet Protocol TV (IPTV) is considered to be the next killer application. With two-way communication capability and point-to-point distribution, each viewer can choose individual programs flexibly and at lower cost. According to a study by the Multimedia Research Group, the number of IPTV subscribers will grow from 4.3 million in 2005 to an estimated 36.8 million by 2009, with a compound annual growth rate of 72 percent, and the IPTV service revenue will increase from $740 million in 2005 to $4.6 billion in 2009, with a compound annual growth rate of 56 percent. Telecommunications service providers are racing to provide IPTV/video on demand (VoD), voice, and data -the so called triple-play services.
A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. I... more A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. In this paper, we give a classification of all extremal double circulant self-dual codes of lengths 74-88.
In this paper it is shown that the weight enumerator of a bordered double circulant self-dual cod... more In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44, 22, double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.
Previously, Type II codes over F4 have been introduced as Euclidean self-dual codes with the prop... more Previously, Type II codes over F4 have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence
I min (2m + 1, m + n + l), is given by we get '3, when n = 0, r = m + 1 , 3 x 2 " -' , w h e n n ... more I min (2m + 1, m + n + l), is given by we get '3, when n = 0, r = m + 1 , 3 x 2 " -' , w h e n n r l , r = m + l , 3 x 22n-1, w h e n n r l , r = m + n + l , , w h e n n r 1 , m + 1 < r 3 2 n + r -m -3 5 min ( 2 m + 1 , m + n ) . d + 1 d i = 1 I = 1 N = 2'N0(d + 1 -I ) = 3dd+' + 1 3 x 2
Conway and Sloane (1990) have listed the possible weight enumerators for extremal self-dual codes... more Conway and Sloane (1990) have listed the possible weight enumerators for extremal self-dual codes up to length 72. In this correspondence, we construct extremal singly-even self-dual [60,30,12] codes whose weight enumerator does not appear in this list. In addition, we present the possible weight enumerators for extremal self-dual codes of length 60
A number of misconceptions and ill-suited approaches concerning fast frequency-hopped (FH) M-ary ... more A number of misconceptions and ill-suited approaches concerning fast frequency-hopped (FH) M-ary NCFSK have risen that have led to the concealment of the impressive performance enhancement capability of some diversity combining methods in the presence of strong intelligent jamming, including partial-band noise (PBN) and multiple tone (MT). Contrary to the conventional wisdom, practical fast FH for anti-jam (AJ) applications uses a constant hop rate whereas the data rate is backed off as needed to provide diversity capability. Then, there is no optimum diversity and the energy per hop is a better measure than the traditional energy per bit used in nonjammed systems. It is then possible to operate under much higher jamming levels than with just the basic FH and standard error-correction techniques alone. These diversity combining techniques are relatively easy to implement, and have been implemented, in real time. Processing gain is an inappropriate measure of performance when applied to non-Gaussian jamming. The perception of a noncoherent combining loss for diversity combining is shown not only to be false, but, for the types of jamming considered, diversity combining actually performs better than coherent combining. Side information is shown to be of little or no value for the types of jamming considered. For large jamming levels, PBN jamming generally causes more degradation in performance than MT jamming
up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search... more up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search or linear programming, or are those which meet the upper bound in [8]. These results are further evidence that the class of QC codes is good, especially in light of the fact that we have only considered 1-generator QC codes.
The class of Quasi-Cyclic Error Correcting Codes is investigated. It is shown that they contain m... more The class of Quasi-Cyclic Error Correcting Codes is investigated. It is shown that they contain many of the best known binary and nonbinary codes. Tables of rate 1/p and (p − 1)/p Quasi-Cyclic (QC) codes are constructed, which are a compilation of previously best known codes as well as many new codes constructed using exhaustive, and other more sophisticated search techniques. Many of these binary codes attain the known bounds on the maximum possible minimum distance, and 13 improve the bounds. The minimum distances and generator polynomials of all known best codes are given. The search methods are outlined and the weight divisibility of the codes is noted. The weight distributions of some s-th Power Residue (PR) codes and related rate 1/s QC codes are found using the link established between PR codes and QC codes. Subcodes of the PR codes are found by deleting certain circulant matrices in the corresponding QC code. They are used as a starting set of circulants for other techniques. Nonbinary Power Residue codes and related QC codes are constructed over GF(3), GF(4), GF(5), GF(7) and GF(8). Their subcodes are also used to find good nonbinary QC codes. A simple and efficient algorithm for constructing primitive polynomials with linearly independent roots over the Galois Field of q elements, GF(q), is developed. Tables of these polynomials are presented. These Tables are unknown for polynomials with nonbinary coefficients, and the known binary Tables are incomplete. The polynomials are employed in such diverse areas as construction of error correcting codes, efficient VLSI implementation of multiplication and inverse operations over Galois Fields, and digital testing of integrated circuits. Using the link established between generalized tail biting convolutional codes and binary QC codes, good QC codes are constructed based on iii Optimum Distance Profile (ODP) convolutional codes. Several best rate 2/3 systematic codes up to circulant size 20 are constructed in this manner.
IEEE Journal on Selected Areas in Communications, 1993
research interests focus on microstrip patch antennas and arrays, monolithic mircrowave integrate... more research interests focus on microstrip patch antennas and arrays, monolithic mircrowave integrated circuits, and code synchronizers for radar and spreadspectrum systems. He has acted for years as a consultant to CAL Corporation, and has conducted many joint research programs with the Communications Research Centre and the Defence Research Establishment Ottawa over the past 16 years.
Let [n; k; d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d ov... more Let [n; k; d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF (q). The following quasicyclic codes are constructed in this paper:
Formally self-dual even codes have recently been studied. Double circulant even codes are a famil... more Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes.
up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search... more up to 18. The best codes, with maximum possible minimum distance, were found by exhaustive search or linear programming, or are those which meet the upper bound in [8]. These results are further evidence that the class of QC codes is good, especially in light of the fact that we have only considered 1-generator QC codes.
Given a quasi-cyclic code with minimum Hamming distance d, a set of convolutional codes is derive... more Given a quasi-cyclic code with minimum Hamming distance d, a set of convolutional codes is derived with free distance equal to d. It is shown that an increase in the rate of these codes results in a decrease in the memory length. The connection between these codes is illustrated with several well-known quasi-cyclic codes. The free distance of some partial unit memory convolutional codes can be determined using the results in this correspondence.
A mobile ad hoc network (MANET) is a collection of mobile nodes that form a wireless network with... more A mobile ad hoc network (MANET) is a collection of mobile nodes that form a wireless network without the use of a fixed infrastructure or centralized administration, and every node acts as a host as well as a router. The topology of an ad hoc network changes frequently and unpredictably. The mobile nature and dynamic topology of MANETs make it very difficult to provide Quality-of-Service (QoS) assurance in such networks. Considering the limited bandwidth and battery power, finding routes that satisfy the bandwidth constraint of applications is a significant challenge.
In this letter, we propose a new modulation scheme called pulse position amplitude modulation (PP... more In this letter, we propose a new modulation scheme called pulse position amplitude modulation (PPAM) for ultra-wideband (UWB) communication systems. PPAM combines pulse position modulation and pulse amplitude modulation to provide good system performance and low computational complexity. The channel capacity of PPAM is determined for a time-hopping multiple-access UWB communication system. The error probability and performance bounds are derived
IEEE Transactions on Wireless Communications, 2005
The capacity and error probability of orthogonal space-time block codes (STBCs) are considered fo... more The capacity and error probability of orthogonal space-time block codes (STBCs) are considered for pulse-amplitude modulation/phase shift keying/quadrature-amplitude modulation (PAM/PSK/QAM) in fading channels. The approach is based on an equivalent scalar additive white Gaussian noise channel with a channel gain proportional to the Frobenius norm of the matrix channel for the STBC. Using this effective channel, capacity and probability of error expressions are derived for PSK/PAM/QAM modulation with space-time block coding. Rayleigh-, Ricean-, and Nakagami-fading channels are considered. As an application, these results are extended to obtain the capacity and probability of error for a multiuser direct sequence code-division multiple-access system employing space-time block coding.
The class of quasi-twisted (QT) codes is a generalization of the class of quasi-cyclic codes, sim... more The class of quasi-twisted (QT) codes is a generalization of the class of quasi-cyclic codes, similar to the way constacyclic codes are a generalization of cyclic codes. In this paper, rate 1/p QT codes over GF(3) are presented which have been constructed using integer linear programming and heuristic combinatorial optimization. Many of these attain the maximum possible minimum distance for any linear code with the given parameters, and several improve the maximum known minimum distances. Two of these new codes, namely (90, 6, 57) and (120, 6, 78), are optimal and so prove that d3(90, 6)=57 and d3(120, 6)=78
Previous results have shown that the class of quasi-cyclic (QC) codes contains many good codes. I... more Previous results have shown that the class of quasi-cyclic (QC) codes contains many good codes. In this paper, new rate (m-1)/pm QC codes over GF(3) and GF(4) are presented. These codes have been constructed using integer linear programming and a heuristic combinatorial optimization algorithm based on a greedy local search. Most of these codes attain the maximum possible minimum distance for any linear code with the same parameters, i.e., they are optimal, and 58 improve the maximum known distances. The generator polynomials for these 58 codes are tabulated, and the minimum distances of rate (m-1)/pm QC codes are given.
nternet Protocol TV (IPTV) is considered to be the next killer application. With two-way communic... more nternet Protocol TV (IPTV) is considered to be the next killer application. With two-way communication capability and point-to-point distribution, each viewer can choose individual programs flexibly and at lower cost. According to a study by the Multimedia Research Group, the number of IPTV subscribers will grow from 4.3 million in 2005 to an estimated 36.8 million by 2009, with a compound annual growth rate of 72 percent, and the IPTV service revenue will increase from $740 million in 2005 to $4.6 billion in 2009, with a compound annual growth rate of 56 percent. Telecommunications service providers are racing to provide IPTV/video on demand (VoD), voice, and data -the so called triple-play services.
A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. I... more A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. In this paper, we give a classification of all extremal double circulant self-dual codes of lengths 74-88.
In this paper it is shown that the weight enumerator of a bordered double circulant self-dual cod... more In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44, 22, double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.
Previously, Type II codes over F4 have been introduced as Euclidean self-dual codes with the prop... more Previously, Type II codes over F4 have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence
I min (2m + 1, m + n + l), is given by we get '3, when n = 0, r = m + 1 , 3 x 2 " -' , w h e n n ... more I min (2m + 1, m + n + l), is given by we get '3, when n = 0, r = m + 1 , 3 x 2 " -' , w h e n n r l , r = m + l , 3 x 22n-1, w h e n n r l , r = m + n + l , , w h e n n r 1 , m + 1 < r 3 2 n + r -m -3 5 min ( 2 m + 1 , m + n ) . d + 1 d i = 1 I = 1 N = 2'N0(d + 1 -I ) = 3dd+' + 1 3 x 2
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Papers by Aaron Gulliver