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Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
DESIGNS FROM LINEAR CODES
Second Edition
Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
ISBN 978-981-125-132-0 (hardcover)
ISBN 978-981-125-133-7 (ebook for institutions)
ISBN 978-981-125-134-4 (ebook for individuals)
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Preface
Sincethepublicationofthefirsteditionofthismonograph,thefollowingadvances
onconstructingcombinatorialt-designswithlinearcodeshavebeenmade:
(cid:15) A generalization of the Assmus-Mattson theorem for linear codes over
finitefields[Tang,DingandXiong(2019)].
(cid:15) ThediscoveryofaninfinitefamilyofnearMDScodesoverfinitefields
supportinganinfinitefamilyof2-designs[DingandTang(2020)].
(cid:15) The discovery of an infinite family of BCH codes over GF(22m+1) of
length 22m+1+1 supporting an infinite family of 4-designs [Tang and
Ding(2021)].
Theseadvancesarethemainmotivationforthesecondeditionofthismonograph.
The Assmus-Mattson theorem was developed in 1969, and has been used to
construct many infinite families of 2-designs and 3-designs in the past 50 years.
However, some infinite families of linear codes do support t-designs, but the t-
designpropertyoftheincidencestructuresdefinedbythelinearcodescannotbe
proved with the Assmus-Mattson theorem and the automorphism group of the
codes. In the past 50 years, a strengthening of the Assmus-Mattson theorem for
special binary linear codes was documented in Calderbank, Delsarte and Sloane
(1991) and several analogues of the Assmus-Mattson theorem in other contexts
(for example, certain association schemes [Morales and Tanaka (2018)], codes
over Z [Tanabe (2000)], and rank-metric codes [Byrne and Ravagnani (2019)])
4
were developed. The Assmus-Mattson theorem for matroids developedin Britz,
RoyleandShiromoto(2009)doescontaintheoriginalAssmus-Mattsontheorem
asaspecialcase,butitbecomestheAssmus-Mattsontheoremwhenitisapplied
to codes. The generalized Assmus-Mattson theorem for linear codes over finite
fieldsdevelopedinTang,DingandXiong(2019)turnsouttobeuseful,andwill
bethetopicofanewchapterinthissecondedition.
v
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vi DesignsfromLinearCodes
It is well-known that MDS codes support only trivial designs (i.e., complete
designs), which are not interesting. The Golay ternary code with parameters
[11,6,5] is near MDS and supports a 4-design. It had been a 70-year-old open
questionwhetherthereisaninfinitefamilyofnearMDScodesoverfinitefields
supporting an infinite family of 2-designs. Similarly, it was open for 71 years
whetherthereisaninfinitefamilyoflinearcodessupportinganinfinitefamilyof
4-designs.Thebreakthroughsregardingthesetwoopenproblemswillbethemain
topicofanothernewchapterinthissecondedition.
Inadditiontothebreakthroughsmentionedabove,otherprogressesondesigns
from linear codes have been also made in the past three years. Based on these
progresses, some chapters in the first addition were revised and new references
wereadded.
Inthissecondedition,anewappendix(namedAppendixC)hasbeenadded.
AppendixCgivesaquickintroductiontoelementarynumbertheory,groups,rings
andfinitefields,andprovidesalotofexercisesonthesetopics. Postgraduatesand
advancedundergraduatesareadvisedtoreadthematerialscarefullyandsolvethe
problems in Appendix C before reading other parts of this book. Another new
appendix (named Appendix A) was also added for reporting some new sporadic
4-designsand5-designssupportedbysomesporadiclinearcodes. Inthissecond
edition, notes are added at the end of most of the chapters for providing further
informationonthetopicscoveredinthechapters.
WeareverygratefultoDr. YanHongNg,Dr. K.K.PhuaandMs. KimTan
ofWorldScientificforhelpinguswiththepublicationofthissecondedition. We
acknowledge the financial support of the Hong Kong Research Grants Council,
underProj. No. 16300418.
CunshengDing
ChunmingTang
Fall2021
November17,2021 14:14 ws-book9x6 DesignsfromLinearCodes designscodes pagevii
Preface to the First Edition
Linear codes andt-designs are companions. On one hand, the incidence matrix
of a t-design generates a linear code over any finite field GF(q). On the other
hand,thesupportsofcodewordsofafixedHammingweightinacodemayforma
t-designundercertainconditions. Interplaybetweencodingtheoryandthetheory
of t-designs has been a very interesting topic for combinatorialists and coding
theorists,andhasbeentreatedtosomeextentinafewmonographsandtextbooks
on coding theory and combinatorics. The purpose of this monograph is to give
a comprehensive treatment of t-designs from linear codes. A special feature of
thismonographistheattentiontothedeterminationoftheparametersoft-designs
heldinlinearcodes.
Thedeterminationoftheweightdistributionsoflinearcodesisanextremely
difficultproblemingeneral,andismuchmoredifficultthanthatoftheminimum
weights of linear codes. The settlement of the parameters oft-designs held in a
linear code over GF(q) is even harder when q>2. A strong motivation of this
monographistherecentadvanceinthedeterminationoftheweightdistributions
of many families of linear codes. This monograph is a strong demonstration of
theusefulnessoftheweightdistributionsoflinearcodes.
Onemayquestionthemotivationofexplicitlyconstructingdesignswithsmall
strength. Ourinterestinexplicitdesignswithsmallstrengthcomesfromthefact
thattheymayyieldlinearcodeswithverygoodparameters,whichcouldbevery
attractive in both theory and practice, although this monograph deals little with
thelinearcodesofdesigns.
Thismonographstudiest-designsmainlyfromlinearcodesoverfinitefields.
A few families of nonlinear codes do yield 3-designs. They include Goethals
codes, Kerdock codes, and some extended perfect codes. The reader is referred
toTonchev(2007)forfurtherinformationondesignsfromthesefamiliesofnon-
linearcodes. Severalfamiliesoft-designshavebeenconstructedfromcodesover
vii
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viii DesignsfromLinearCodes
Z . InformationaboutthesedesignscouldbefoundinHelleseth,RongandYang
4
(2001). Thesearethedesignsmissingfromthisbook.
Inthismonograph,wepaymuchmoreattentiontoinfinitefamiliesofdesigns
thansporadicdesigns. Therefore,wewillnottreatMathieugroups,whicharethe
automorphismgroupsofsomelinearcodesandt-designs.
This book is intended to be a reference for postgraduates and researchers in
theareasofcombinatorics,codingtheoryandcommunicationsengineering. The
readerisassumedtohavethebasicknowledgeoflinearalgebraandfinitefields.
Nevertheless, some mathematical foundations are recalled and summarized in
Chapter 1. In many places, proofs of some results are omitted. In this case, a
referenceofsuchresultisgiven,sothatthereadercouldfindaproof.
I am indebted to Professor Vladimir D. Tonchev for beneficial discussions
on interplay betweent-designs and linear codes. I am grateful to my co-authors
for collaborations on coding theory, combinatorics and cryptography as well as
relatedtopics. IthankXiranAi,Dr. DonMak,Dr. K.K.Phua,andNingTuatthe
WordScientificfortheirassistancewiththepublicationofthismonograph.
CunshengDing
HongKong
Spring2018
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Contents
Preface v
PrefacetotheFirstEdition vii
1. MathematicalFoundations 1
1.1 TheRingsZ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
n
1.2 FiniteFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 IntroductiontoFiniteFields . . . . . . . . . . . . . . . 2
1.2.2 Traces,Norms,andBases . . . . . . . . . . . . . . . . 3
1.2.3 FieldAutomorphisms . . . . . . . . . . . . . . . . . . 4
1.2.4 AdditiveandMultiplicativeCharacters . . . . . . . . . 4
1.2.5 SeveralTypesofCharacterSums . . . . . . . . . . . . 6
1.2.6 QuadraticFormsoverGF(q) . . . . . . . . . . . . . . 9
1.3 GroupAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 SpecialTypesofPolynomials . . . . . . . . . . . . . . . . . . . 11
1.4.1 PermutationPolynomialsoverFiniteFields . . . . . . . 11
1.4.2 DicksonPolynomialsoverFiniteFields . . . . . . . . . 12
1.4.3 KrawtchoukPolynomials . . . . . . . . . . . . . . . . 13
1.5 CyclotomyinGF(r) . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 Cyclotomy . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.2 CyclotomyinGF(r) . . . . . . . . . . . . . . . . . . . 18
1.6 FiniteGeometries . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.1 ProjectiveSpacesPG(m,GF(q)) . . . . . . . . . . . . 21
1.6.2 AffineSpacesAG(m,GF(q)) . . . . . . . . . . . . . . 23
1.6.3 ProjectivePlanes . . . . . . . . . . . . . . . . . . . . . 23
1.6.4 DesarguesianProjectivePlanesPG(2,GF(q)). . . . . . 25
ix
