A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over \ZZ_4, the integers $\bmod~4$. The Kerdock and Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over \ZZ_4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in general are not, nor is the Golay code.Comment: 5 page

Low Correlation Sequences over the QAM Constellation

This paper presents the first concerted look at low correlation sequence families over QAM constellations of size M^2=4^m and their potential applicability as spreading sequences in a CDMA setting. Five constructions are presented, and it is shown how such sequence families have the ability to transport a larger amount of data as well as enable variable-rate signalling on the reverse link. Canonical family CQ has period N, normalized maximum-correlation parameter theta_max bounded above by A sqrt(N), where 'A' ranges from 1.8 in the 16-QAM case to 3.0 for large M. In a CDMA setting, each user is enabled to transfer 2m bits of data per period of the spreading sequence which can be increased to 3m bits of data by halving the size of the sequence family. The technique used to construct CQ is easily extended to produce larger sequence families and an example is provided. Selected family SQ has a lower value of theta_max but permits only (m+1)-bit data modulation. The interleaved 16-QAM sequence family IQ has theta_max <= sqrt(2) sqrt(N) and supports 3-bit data modulation. The remaining two families are over a quadrature-PAM (Q-PAM) subset of size 2M of the M^2-QAM constellation. Family P has a lower value of theta_max in comparison with Family SQ, while still permitting (m+1)-bit data modulation. Interleaved family IP, over the 8-ary Q-PAM constellation, permits 3-bit data modulation and interestingly, achieves the Welch lower bound on theta_max.Comment: 21 pages, 3 figures. To appear in IEEE Transactions on Information Theory in February 200

Single and multi-antenna MC-DS-CDMA with joint detection for broadband block-fading channels

In the context of broadband wireless communications using code division multiple access (CDMA), the main multiple access (MA) options include single-carrier direct sequence CDMA (SC-DS-CDMA) using time-domain direct sequence spreading [1, p. 728], multicarrier CDMA (MC-CDMA) using frequency-domain spreading [2, 3] and multicarrier DS-CDMA (MC-DS-CDMA) using time-domain direct sequence spreading of the individual sub-carrier signals [4, 5]. It was shown in [6] that MC-DS-CDMA has the highest degrees of freedom in the family of CDMA schemes that can be beneficially exploited during the system design and reconfiguration procedures. An amalgam of MC-CDMA and MC-DS-CDMA known as time and frequency domain spreading (TF-domain spreading) MC-DS-CDMA was proposed in [6]. TF-domain spreading MC-DS-CDMA has several benefits over conventional MC-DS-CDMA with regard to both capacity and performance [7]. However, in contrast to conventional MC-DS-CDMA, TF-domain spreading MC-DS-CDMA introduces MUI, which necessitates the use of joint detection at the receiver. Recently, multiple input multiple output (MIMO) or multi-antenna TF-domain spreading MC-DS-CDMA schemes have been proposed in the literature that e ciently exploit both the spatial and frequency diversity available in MIMO frequency-selective channels [8, 9]. Although an extensive amount of research has been done on single and multi-antenna TF-domain spreading MC-DS-CDMA schemes that achieve both spatial and frequency diversity in frequency-selective slow fading channels [6–9], very little research considers the time-selectivity of the wireless channels encountered. Thus, the above-mentioned schemes may not be su ciently e cient, when communicating over wireless channels exhibiting both frequency-selective and time-selective fading. There are very few MC-DS-CDMA schemes in the literature that consider the time-selectivity of the wireless channels encountered. This study considers the design of single and multi-antenna TF-domain spreading MC-DS-CDMA, for frequency-selective block-fading channels, which are capable of exploiting the full diversity available in the channel (i.e. spatial, frequency and temporal diversity), using various methods of joint detection at the receiver. It has been shown that the diversity gain in block-fading channels can be improved by coding across multiple fading blocks [10–12]. Single-antenna TF-domain spreading MC-DS-CDMA is considered for the quasi-synchronous uplink channel, and multi-antenna TF-domain spreading MC-DS-CDMA is considered for the synchronous downlink channel. Numerous simulated bit error rate (BER) performance curves, obtained using a triply selective MIMO channel platform, are presented in this study using optimal and sub-optimal joint detection algorithms at the receiver. In addition, this study investigates the impact of spatial correlation on the BER performance of the MC-DS-CDMA schemes considered. From these simulated results, one is able to conclude that TF-domain spreading MC-DS-CDMA designed for frequency-selective block-fading channels performs better than previously proposed schemes designed for frequency-selective slow fading channels, owing to the additional temporal diversity exploited under the block-fading assumption. AFRIKAANS : In die konteks van bre¨eband- draadlose kommunikasie deur die gebruik van kodeverdelingveelvuldige toegang (KVVT) behels die belangrikste veelvuldigetoegang- (VT) opsies enkel-draer direkte-sekwensie KVVT (ED-DS-KVVT), deur die gebruik van tyd-domein direkte sekwensie-verspreiding [1, p. 728], veelvuldigedraer-KVVT (VD-KVVT) deur die gebruik van frekwensiedomein-verspreiding [2, 3] en VD-DS- KVVT deur die gebruik van tyd-domein direkte sekwensie-verspreiding van die individuele sub-draerseine [4, 5]. Daar is in [6] aangetoon dat VD-DS-KVVT die hoogste vlakke van vryheid in die familie KVVT-skemas het wat voordelig benut kan word gedurende sisteemontwerp en rekonfigurasieprosedures. ’n Amalgaam van VD-KVVT en VD-DS-KVVT bekend as tyd-en-frekwensiedomeinverspreiding (TF-domeinverspreiding) VD-DS-KVVT is voorgestel in [6]. TF-domeinverspreiding VD-DS-KVVT het verskeie voordele bo konvensionele VD-DS-KVVT wat sowel kapasiteit as werkverrigting betref [7]. In teenstelling met konvensionele VD-DS-KVVT benut TF-domeinverspreiding VD-DS-KVVT multi-gebruiker-interferensie, wat die gebruik van gesamentlike opsporing by die ontvanger noodsaak. In die onlangse verlede is in die literatuur veelvuldige-inset-veelvuldige-uitset- (VIVU) of veelvuldige-antenna TF-omeinverspreiding VD-DS-KVVT-skemas voorgestel wat sowel die ruimtelike as frekwensiediversiteit wat in VIVU frekwensie-selektiewe kanale beskikbaar is, e ektief gebruik [8, 9]. Hoewel uitgebreide navorsing onderneem is oor enkel- en multi-antenna TF-domeinverspreiding VD-DS-KVVT-skemas wat sowel ruimtelike as frekwensie diversiteit in frekwensie-selektiewe stadig deinende kanale bereik [6–9], oorweeg baie min navorsing die tyd-selektiwiteit van die draadlose kanale wat betrokke is. Bogenoemde skemas mag dus nie e ektief genoeg wees nie wanneer kommunikasie plaasvind oor draadlose kanale wat sowel frekwensie-selektiewe as tyd-selektiewe wegsterwing toon. Baie min VD-DS-KVVT-skemas in die literatuur skenk aandag aan die tyd-selektiwiteit van die betrokke draadlose kanale. Die studie ondersoek die ontwerp van enkel- en multi-antenna TF-domeinverspreiding VD-DS-KVVT vir frekwensie-selektiewe blokwegsterwingkanale, wat in staat is om die volle diversiteit wat in die kanaal beskikbaar is, te benut (i.e. ruimtelike, frekwensie- en tyddiversiteit), deur die gebruik van verskeie metodes van gesamentlike opsporing by die ontvanger. Daar is aangetoon dat die diversiteitwins in blokwegsterwingkanale verbeter kan word deur kodering oor veelvuldige deinende blokke [10–12]. Enkel-antenna TF-domeinverspreiding VD-DS-KVVT word oorweeg vir die kwasi-sinchroniese opverbinding-kanaal, en multi-antenna TF-domeinverspreiding VD-DS-KVVT vir die sinchroniese afverbinding-kanaal. Talryke gesimuleerde bisfouttempo (BFT) werkverrigtingkurwes wat verkry is deur die gebruik van ’n drie-voudige selektiewe VIVU-kanaalplatform, word in hierdie studie aangebied, deur die gebruik van optimale en sub-optimale gesamentlike opsporingsalgoritmes by die ontvanger. Daarbenewens ondersoek hierdie studie die impak van ruimtelike korrelasie op die BFT-werkverrigring van die VD-DS-KVVT-skemas wat oorweeg word. Uit hierdie gesimuleerde resultate is dit moontlik om tot die gevolgtrekking te kom dat TF-domeinverspreiding VD-DS-KVVT wat ontwerp is vir frekwensie-selektiese blokwegsterwingkanale beter werkverrigting toon as vroe¨er voorgestelde skemas wat ontwerp is vir frekwensie-selektiewe stadig deinende kanale, te danke aan die ekstra tyddiversiteit wat deur die blokwegsterwing-aanname benut word. CopyrightDissertation (MEng)--University of Pretoria, 2010.Electrical, Electronic and Computer Engineeringunrestricte

Code design and analysis for multiple access communications

This thesis explores various coding aspects of multiple access communications, mainly for spread spectrum multiaccess(SSMA) communications and collaborative coding multiaccess(CCMA) communications. Both the SSMA and CCMA techniques permit efficient simultaneous transmission by several users sharing a common channel, without subdivision in time or frequency. The general principle behind these two multiaccess schemes is that one can find sets of signals (codes) which can be combined together to form a composite signal; on reception, the individual signals in the set can each be recovered from the composite signal. For the CCMA scheme, the isolation between users is based on the code structure; for the SSMA scheme, on the other hand, the isolation between users is based on the autocorrelation functions(ACFs) and crosscorrelation functions (CCFs) of the code sequences. It is clear that, in either case, the code design is the key to the system design.For the CCMA system with a multiaccess binary adder channel, a class of superimposed codes is analyzed. It is proved that every constant weight code of weight w and maximal correlation λ corresponds to a subclass of disjunctive codes of order T 3, the out-of-phase ACFs and CCFs of the codes are constant and equal to √L. In addition, all codes of the same length are mutually orthogonal.2. Maximal length sequences (m-sequences) over Gaussian integers, suitable for use with QAM modulation, are considered. Two sub-classes of m-sequences with quasi-perfect periodic autocorrelations are obtained. The CCFs between the decimated m-sequences are studied. By applying a simple operation, it is shown that some m-sequences over rational and Gaussian integers can be transformed into perfect sequences with impulsive ACFs.3. Frank codes and Chu codes have perfect periodic ACFs and optimum periodic CCFs. In addition, it is shown that they also have very favourable nonperiodic ACFs; some new results concerning the behaviour of the nonperiodic ACFs are derived. Further, it is proved that the sets of combinedFrank/Chu codes, which contain a larger number of codes than either of the two constituent sets, also have very good periodic CCFs. Based on Frank codes and Chu codes, two interesting classes of real-valued codes with good correlation properties are defined. It is shown that these codes have periodic complementary properties and good periodic and nonperiodic ACF/CCFs.Finally, a hybrid CCMA/SSMA coding scheme is proposed. This new hybrid coding scheme provides a very flexible and powerful multiple accessing capability and allows simple and efficient decoding. Given an SSMA system with K users and a CCMA system with N users, where at most T users are active at any time, then the hybrid system will have K . N users with at most T.K users active at any time. The hybrid CCMA/SSMA coding scheme is superior to the individual CCMA system or SSMA system in terms of information rate, number of users, decoding complexity and external interference rejection capability

Three-weight codes over rings and strongly walk regular graphs

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over $\mathbb{Z}_{p^m},$ for $p$ a prime, and more generally over chain rings of depth $m$, and with a residue field of size $q$, a prime power. Infinite families of examples are built from Kerdock and generalized Teichm\"uller codes. As a byproduct, we give an alternative proof that the Kerdock code is nonlinear.Comment: 28 pages, 6 table

Code design and analysis for multiple access communications

This thesis explores various coding aspects of multiple access communications, mainly for spread spectrum multiaccess(SSMA) communications and collaborative coding multiaccess(CCMA) communications. Both the SSMA and CCMA techniques permit efficient simultaneous transmission by several users sharing a common channel, without subdivision in time or frequency. The general principle behind these two multiaccess schemes is that one can find sets of signals (codes) which can be combined together to form a composite signal; on reception, the individual signals in the set can each be recovered from the composite signal. For the CCMA scheme, the isolation between users is based on the code structure; for the SSMA scheme, on the other hand, the isolation between users is based on the autocorrelation functions(ACFs) and crosscorrelation functions (CCFs) of the code sequences. It is clear that, in either case, the code design is the key to the system design.For the CCMA system with a multiaccess binary adder channel, a class of superimposed codes is analyzed. It is proved that every constant weight code of weight w and maximal correlation λ corresponds to a subclass of disjunctive codes of order T 3, the out-of-phase ACFs and CCFs of the codes are constant and equal to √L. In addition, all codes of the same length are mutually orthogonal.2. Maximal length sequences (m-sequences) over Gaussian integers, suitable for use with QAM modulation, are considered. Two sub-classes of m-sequences with quasi-perfect periodic autocorrelations are obtained. The CCFs between the decimated m-sequences are studied. By applying a simple operation, it is shown that some m-sequences over rational and Gaussian integers can be transformed into perfect sequences with impulsive ACFs.3. Frank codes and Chu codes have perfect periodic ACFs and optimum periodic CCFs. In addition, it is shown that they also have very favourable nonperiodic ACFs; some new results concerning the behaviour of the nonperiodic ACFs are derived. Further, it is proved that the sets of combinedFrank/Chu codes, which contain a larger number of codes than either of the two constituent sets, also have very good periodic CCFs. Based on Frank codes and Chu codes, two interesting classes of real-valued codes with good correlation properties are defined. It is shown that these codes have periodic complementary properties and good periodic and nonperiodic ACF/CCFs.Finally, a hybrid CCMA/SSMA coding scheme is proposed. This new hybrid coding scheme provides a very flexible and powerful multiple accessing capability and allows simple and efficient decoding. Given an SSMA system with K users and a CCMA system with N users, where at most T users are active at any time, then the hybrid system will have K . N users with at most T.K users active at any time. The hybrid CCMA/SSMA coding scheme is superior to the individual CCMA system or SSMA system in terms of information rate, number of users, decoding complexity and external interference rejection capability

Generalizations of Bent Functions. A Survey

Bent functions (Boolean functions with extreme nonlinearity properties) are actively studied for their numerous applications in cryptography, coding theory, and other fields. New statements of problems lead to a large number of generalizations of the bent functions many of which remain little known to the experts in Boolean functions. In this article, we offer a systematic survey of them