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Construction of quantum caps in projective space PG(r, 4) and quantum codes of distance 4

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Abstract

Constructions of quantum caps in projective space PG(r, 4) by recursive methods and computer search are discussed. For each even n satisfying \(n\ge 282\) and each odd z satisfying \(z\ge 275\), a quantum n-cap and a quantum z-cap in \(PG(k-1, 4)\) with suitable k are constructed, and \([[n,n-2k,4]]\) and \([[z,z-2k,4]]\) quantum codes are derived from the constructed quantum n-cap and z-cap, respectively. For \(n\ge 282\) and \(n\ne 286\), 756 and 5040, or \(z\ge 275\), the results on the sizes of quantum caps and quantum codes are new, and all the obtained quantum codes are optimal codes according to the quantum Hamming bound. While constructing quantum caps, we also obtain many large caps in PG(r, 4) for \(r\ge 11\). These results concerning large caps provide improved lower bounds on the maximal sizes of caps in PG(r, 4) for \(r\ge 11\).

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Acknowledgments

This work is supported by National Natural Science Foundations of China under Grant Nos. 11471011, 11071255 and “973” program of China under Grant No. 2013CB834204. A part of this work was carried out while R. Li was visiting the Chern Institute of Mathematics (CIM) at Nankai University. R. Li is grateful to the Chern Institute for kind hospitality.

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Authors and Affiliations

  1. School of Science, Air Force Engineering University, Xi’an, 710051, Shaanxi, People’s Republic of China

    Ruihu Li, Qiang Fu & Luobin Guo

  2. Center for Combinatorics, Nankai University, Tianjin, 300071, People’s Republic of China

    Xueliang Li

Authors
  1. Ruihu Li
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  2. Qiang Fu
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  3. Luobin Guo
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  4. Xueliang Li
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Corresponding author

Correspondence to Ruihu Li.

Appendices

Appendix A

The 126-cap \(\mathbf {G}_{6, 126}\) mentioned in Sect. 2 is as follows:

$$\begin{aligned} \mathbf {G}_{6,126}=(\mathbf {Q}_{1}, \mathbf {Q}_{2},\ldots , \mathbf {Q}_{15},\mathbf { R})=(\mathbf {M}_{1}\quad \mathbf {M}_{2}\quad \mathbf {M}_{3}), \end{aligned}$$

where

$$\begin{aligned} \mathbf {M}_1= & {} (\mathbf {Q}_{1},\mathbf {Q}_{2},\ldots ,\mathbf {Q}_{6})\\&=\left( \begin{array}{c} 111111111111111111111111 111111111111111111111111\\ 231223121312121313232313 002312310023123100322113\\ 000011330022112233332211 232323231212121213131313\\ 112211223322223333111133 001122330011223300112233\\ 000000000000000000000000 111111112222222233333333\\ 000011330011222211223333 000000000000000000000000\\ \end{array} \right) ,\\ \mathbf {M}_2= & {} (\mathbf {Q}_{7},\mathbf {Q}_{8},\ldots ,\mathbf {Q}_{12})\\&=\left( \begin{array}{c} 111111111111 111111111111 111111111111111111111111\\ 112321132232 213133232131 333212131123211322231213\\ 232323231212 121213131313 232323231212121213131313\\ 001122330011 223300112233 001122330011223300112233\\ 111111112222 222233333333 111111112222222233333333\\ 111111111111 111111111111 222222222222222222222222\\ \end{array} \right) ,\\ \mathbf {M}_3= & {} (\mathbf {Q}_{13},\mathbf {Q}_{14},\mathbf {Q}_{15},\mathbf {R})\\&=\left( \begin{array}{c} 111111111111111111111111 000000\\ 223221313332121311321231 111111\\ 232323231212121213131313 231213\\ 001122330011223300112233 000000\\ 111111112222222233333333 000000\\ 333333333333333333333333 112233\\ \end{array} \right) . \end{aligned}$$

Appendix B

The weight polynomial of code generated by 288-cap \(\mathbf {G}_{7,288}\) in PG(6, 4) is \(Wt_{288}(z)=1+1089z^{202}+270z^{203}+120z^{204}+990z^{206}+18z^{207}+225z^{210} +5400z^{215}+900z^{216}+3267z^{218}+360z^{219}+2970z^{222}+675z^{226} +3z^{256}+90z^{267}+6z^{271}\).

The matrix \(\mathbf {G}'_{7,288}\) given in Sect. 3 is as follows:

$$\begin{aligned} \mathbf {G}'_{7,288} = ({\mathbf {S}'_1}\;{\mathbf {S}'_2}\;{\mathbf {S}'_3}\;{\mathbf {S}'_4}\;{\mathbf {S}'_5}) =\left( \begin{array}{ccc} 1&{} \cdots &{}1\\ {\widetilde{\eta }_1}&{} \cdots &{}{\widetilde{\eta } _{271}} \end{array}\qquad \begin{array}{ccc} 0&{} \cdots &{}0 \\ {\widetilde{\eta }_{272}}&{} \cdots &{}{\widetilde{\eta } _{288}} \end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} {\mathbf {S}'_1}= & {} \left( \begin{array}{c} 111111111111111111111111111111111111111111111111111111111111\\ 333333333333333333333333333333333333333333333333222222222222\\ 000000000000000022222222222222222222222222222222333333333333\\ 000000000000000000000000000000003333333333333333333333333333\\ 011123323133122201112332313312220111233231331222033312212322\\ 002213203102113300221320310211330022132031021133001132102301\\ 000311310110333200031131011033320003113101103332000233230330 \end{array}\right) ,\\ {\mathbf {S}'_2}= & {} \left( \begin{array}{c} 111111111111111111111111111111111111111111111111111111111111\\ 222211111111111111111111111111111111222222222222222211111111\\ 333311111111111111113333333333333333000000000000000011111111\\ 333322222222222222221111111111111111333333333333333300000000\\ 311102223113121123330222311312112333033312212322311102223113\\ 332200332130120322110033213012032211001132102301332200332130\\ 222100012212022011130001221202201113000233230330222100012212 \end{array} \right) , \end{aligned}$$
$$\begin{aligned} {\mathbf {S}'_3}= & {} \left( \begin{array}{c} 111111111111111111111111111111111111111111111111111111111111\\ 111111113333333333333333111111111111111111111111111111113333\\ 111111111111111111111111000000000000000033333333333333331111\\ 000000001111111111111111222222222222222200000000000000003333\\ 121123330111233231331222022231131211233302223113121123330111\\ 120322110022132031021133003321301203221100332130120322110022\\ 022011130003113101103332000122120220111300012212022011130003 \end{array} \right) ,\\ {\mathbf {S}'_4}= & {} \left( \begin{array}{c} 111111111111111111111111111111111111111111111111111111111111\\ 333333333333222222222222222222222222222222222222222222222222\\ 111111111111333333333333333322222222222222222222222222222222\\ 333333333333222222222222222222222222222222221111111111111111\\ 233231331222033312212322311103331221232231110333122123223111\\ 132031021133001132102301332200113210230133220011321023013322\\ 113101103332000233230330222100023323033022210002332303302221 \end{array} \right) ,\\ {\mathbf {S}'_5}= & {} \left( \begin{array}{c} 111111111111111111111111111111100000000000000000\\ 000000000000000000000000000000001111111111111111\\ 000000000000000022313011031322000000000000000000\\ 111111111111111112230210312330000000000000000000\\ 022231131211233322231131211233310222311312112333\\ 003321301203221100000000000000000033213012032211\\ 000122120220111322231131211233310001221202201113 \end{array} \right) . \end{aligned}$$

The weight polynomials of the codes and their dual generated by the quantum caps constructed in Sect. 3.2 are as follows:

$$\begin{aligned} Wt_{256}(z)= & {} 1+288z^{176}+2304z^{180}+6780z^{192}+6912z^{196}+96z^{240}+3z^{256},\\ Wt^{\perp }_{256}(y)= & {} 1+1220160y^{4}+116951040y^{5}+\cdots .\\ Wt_{262}(z)= & {} 1+156z^{180}+180z^{182}+1584z^{184}+672z^{186}+60z^{192}+5184z^{196}\\&+\,1680z^{198}+4752z^{200}+2016z^{202}+36z^{244}+60z^{246}+3z^{256},\\ Wt^{\perp }_{262}(y)= & {} 1+1316685y^{4}+132678720y^{5}+\cdots .\\ Wt_{268}(z)= & {} 1+3z^{180}+54z^{184}+252z^{186}+1179z^{188}+900z^{190}+204z^{192}\\&+\,81z^{196}+48z^{198}+3942z^{200}+2556z^{202}+3753z^{204}+2700z^{206}\\&+\,612z^{208}+72z^{250}+24z^{252}+3z^{256},\\ Wt^{\perp }_{268}(y)= & {} 1+1421013y^{4}+149837328y^{5}+\cdots .\\ Wt_{282}(z)= & {} 1+6z^{194}+150z^{196}+1062z^{198}+897z^{200}+453z^{202}+117z^{204}\\&+\,27z^{206}+408z^{208}+2754z^{210}+2571z^{212}+3609z^{214}+2706z^{216}\\&+\,1200z^{218}+279z^{220}+45z^{222}+3z^{256}+3z^{260}+57z^{262}\\&+\,33z^{264}+3z^{266},\\ Wt^{\perp }_{282}(y)= & {} 1+1698990y^{4}+195873432y^{5}+\cdots .\\ Wt_{284}(z)= & {} 1+12z^{196}+264z^{198}+1332z^{200}+648z^{202}+324z^{204}+120z^{206}\\&+\,48z^{208}+720z^{210}+3252z^{212}+2472z^{214}+4044z^{216}+1680z^{218}\\&+\,1116z^{220}+216z^{222}+36z^{224}+3z^{256}+72z^{264}+24z^{266},\\ Wt^{\perp }_{284}(y)= & {} 1+1742325y^{4}+203275920y^{5}+\cdots .\\ Wt_{286}(z)= & {} 1+1089z^{200}+270z^{202}+1110z^{204}+18z^{206}+225z^{208}+5400z^{214}\\&+\,4167z^{216}+360z^{218}+2970z^{220}+675z^{224}+3z^{256}+90z^{266}+6z^{270},\\ Wt^{\perp }_{286}(y)= & {} 1+1791435y^{4}+210841560y^{5}+\cdots . \end{aligned}$$

Appendix C

At the beginning of Sect. 4, we have pointed out that [13] constructed six large caps in PG(r, 4) for \(7\le r\le 12\). These six large caps are: 756-cap in PG(7, 4), 2136-cap in PG(8, 4), 5124-cap in PG(9, 4), 15840-cap in PG(10, 4), 36084-cap in PG(11, 4) and 95256-cap in PG(12, 4). The weight polynomials of cap codes of these six caps and their dual codes are as follows:

$$\begin{aligned} Wt_{756}(z)= & {} 1+27z^{504}+450z^{516}+432z^{520}+981z^{528}+135z^{552}+6480z^{560}\\&+\,32346z^{564}+7317z^{568}+5679z^{576}+10206z^{580}+1296z^{584}\\&+\,108z^{592}+63z^{720}+9z^{744}+6z^{756},\\ Wt_{756}^{\perp }(y)= & {} 1+25841403y^{4}+\cdots .\\ Wt_{2136}(z)= & {} 1+135z^{1490}+585z^{1492}+2160z^{1500}+45z^{1504}+135z^{1506}+\cdots \\&+\,45z^{1986}+45z^{2002}+3z^{2010}+15z^{2034}.\\ Wt^{\perp }_{2136}(z)= & {} 1+393124770y^{4}+307120772160y^{5}+\cdots ,\\ Wt_{5124}(z)= & {} 1+3z^{2988}+6z^{3030}+3z^{3240}+6z^{3270}+3z^{3276}+3z^{3486}+27z^{3498}\\&+\,6z^{3504}+\cdots +24z^{4710}+3z^{4746}+3z^{4758}+45z^{4782}+15z^{4878}.\\ Wt_{5124}^{\perp }(y)= & {} 1+3378253707y^4+\cdots ,\\ Wt_{15840}(z)= & {} 1+441z^{11016}+270z^{11052}+360z^{11064}+1512z^{11080}+1080z^{11112}\\&+\cdots +27z^{12424}+3z^{14400}+90z^{14892}+30z^{15084},\\ Wt^{\perp }_{15840}(y)= & {} 1+83251482696y^{4}+417595251328704y^{5}+\cdots .\\ Wt_{36084}(z)= & {} 1+135z^{25140}+6z^{25204}+36z^{25206}+180z^{25208}+3z^{25210}+\cdots \\&+9z^{33072}+57z^{33450}+45z^{33812}+3z^{33954}+15z^{34356},\\ Wt^{\perp }_{36084}(y)= & {} 1+564963703299y^{4}+6475053526943088y^{5}+\cdots ,\\ Wt_{95256}(z)= & {} 1+18z^{63576}+378z^{65016}+144z^{65400}+864z^{65880}+3969z^{66240}\\&+\,1890z^{66528}+\cdots +1728z^{73560}+72z^{74136}+144z^{74520}+126z^{90720}\\&+\,9z^{92376}+3z^{95256},\\ Wt_{95256}^{\perp }(y)= & {} 1+8379131756670y^{4}+\cdots . \end{aligned}$$

Appendix D

The weight polynomials of the code generated by 36150-cap \(\mathbf {G}_{12,36150}\) and its dual code in PG(11, 4) are

$$\begin{aligned} Wt_{36150}(z)= & {} 1+135z^{25206}+3z^{25246}+219z^{25252}+3z^{25258}+24z^{25320}+6z^{25326}\\&+\,36z^{25332}+12z^{25338}+39z^{25344}+42z^{25348}+81z^{25350}+54z^{25354}\\&+\,159z^{25356}+60z^{25360}+57z^{25362}+30z^{25366}+24z^{25368}+18z^{25374}\\&+\,123z^{25380}+42z^{25386}+576z^{25388}+39z^{25392}+24z^{25398}+9z^{25404}\\&+\,3z^{25406}+12z^{25410}+39z^{25412}+6z^{25416}+3z^{25418}+30z^{25422}\\&+\,63z^{25428}+54z^{25434}+54z^{25440}+81z^{25446}+111z^{25452}+48z^{25458}\\&+\,60z^{25464}+108z^{25470}+99z^{25476}+144z^{25478}+333z^{25480}\\&+\,549z^{25482}+72z^{25484}+33z^{25488}+27z^{25494}+12z^{25500}+24z^{25502}\\&+\,18z^{25506}+48z^{25508}+144z^{25510}+216z^{25512}+486z^{25514}\\&+\,72z^{25516}+144z^{25518}+18z^{25520}+18z^{25526}+18z^{25532}+12z^{25534}\\&+\,24z^{25538}+24z^{25540}+12z^{25544}+9z^{25546}+9z^{25548}+9z^{25552}\\&+\,18z^{25554}+9z^{25558}+27z^{25560}+9z^{25564}+15z^{25566}+12z^{25570}\\&+\,36z^{25572}+6z^{25576}+21z^{25578}+9z^{25584}+27z^{25590}+6z^{25596}\\&+\,12z^{25602}+9z^{25608}+3z^{25620}+144z^{25638}+144z^{25640}+432z^{25642}\\&+\,72z^{25644}+288z^{25646}+3z^{25668}+72z^{25670}+72z^{25672}+222z^{25674}\\&+\,36z^{25676}+144z^{25678}+6z^{25680}+6z^{25686}+27z^{25692}+12z^{25698}\\&+\,12z^{25724}+36z^{25726}+42z^{25732}+9z^{25734}+6z^{25738}+45z^{25750}\\&+\,6z^{25756}+48z^{25758}+36z^{25764}+3z^{25770}+12z^{25800}+3z^{25806}\\ \end{aligned}$$
$$\begin{aligned}&\,\,\,\qquad \qquad \qquad +\,18z^{25812}+6z^{25818}+72z^{25822}+6z^{25824}+6z^{25830}+45z^{25848}\\&\,\,\,\qquad \qquad \qquad +\,144z^{25854}+81z^{25860}+81z^{25866}+36z^{25868}+48z^{25872}+18z^{25878}\\&\,\,\,\qquad \qquad \qquad +\,60z^{25886}+30z^{25892}+51z^{25902}+81z^{25908}+51z^{25914}\\&\,\,\,\qquad \qquad \qquad +\,30z^{25918}+99z^{25920}+15z^{25924}+93z^{25926}+48z^{25932}+51z^{25938}\\&\,\,\,\qquad \qquad \qquad +\,51z^{25944}+90z^{25950}+42z^{25956}+3z^{25962}+9z^{25968}+72z^{25990}\\&\,\,\,\qquad \qquad \qquad +\,144z^{25992}+252z^{25994}+36z^{25996}+6z^{26018}+3z^{26040}+3z^{26046}\\&\,\,\,\qquad \qquad \qquad +\,3z^{26050}+3z^{26052}+6z^{26236}+18z^{26238}+21z^{26244}+3z^{26250}\\&\,\,\,\qquad \qquad \qquad +\,36z^{26334}+6z^{26352}+33z^{26358}+33z^{26364}+3z^{26370}+21z^{26382}\\&\,\,\,\qquad \qquad \qquad +\,27z^{26388}+18z^{26394}+21z^{26400}+39z^{26406}+15z^{26412}+9z^{26418}\\&\,\,\,\qquad \qquad \qquad +\,3z^{26530}+42z^{26576}+105z^{26582}+93z^{26588}+180z^{26592}+54z^{26594}\\&\,\,\,\qquad \qquad \qquad +\,90z^{26598}+15z^{26600}+585z^{26604}+6z^{26606}+270z^{26610}+180z^{26616}\\&\,\,\,\qquad \qquad \qquad +\,216z^{26640}+144z^{26642}+324z^{26644}+396z^{26646}+54z^{26654}\\&\,\,\,\qquad \qquad \qquad +\,144z^{26656}+576z^{26658}+807z^{26660}+432z^{26662}+396z^{26664}\\&\,\,\,\qquad \qquad \qquad +366z^{26666}+216z^{26668}+99z^{26672}+147z^{26678}+108z^{26684}\\&\,\,\,\qquad \qquad \qquad +\,144z^{26686}+288z^{26688}+489z^{26690}+468z^{26692}+252z^{26694}\\&\,\,\,\qquad \qquad \qquad +\,9z^{26696}+147z^{26704}+417z^{26710}+477z^{26716}+96z^{26720}\\&\,\,\,\qquad \qquad \qquad +\,279z^{26722}+123z^{26726}+153z^{26728}+90z^{26732}+117z^{26734}\\&\,\,\,\qquad \qquad \qquad +\,42z^{26736}+42z^{26738}+87z^{26740}+111z^{26742}+9z^{26744}+42z^{26746}\\&\,\,\,\qquad \qquad \qquad +\,105z^{26748}+24z^{26752}+54z^{26754}+39z^{26758}+87z^{26760}+57z^{26764} \end{aligned}$$
$$\begin{aligned}&+\,111z^{26766}+432z^{26768}+336z^{26770}+726z^{26772}+792z^{26774}\\&+\,711z^{26776}+2466z^{26778}+3276z^{26780}+1956z^{26782}+696z^{26784}\\&+\,1764z^{26786}+2961z^{26788}+1698z^{26790}+2268z^{26792}+5655z^{26794}\\&+\,3492z^{26796}+324z^{26798}+1221z^{26800}+1656z^{26802}+1476z^{26804}\\&+\,1110z^{26806}+432z^{26808}+720z^{26810}+1515z^{26812}+1329z^{26814}\\&+\,1908z^{26816}+3264z^{26818}+3420z^{26820}+1908z^{26822}+1104z^{26824}\\&+\,834z^{26826}+540z^{26828}+822z^{26830}+387z^{26832}+609z^{26836}\\&+\,894z^{26838}+252z^{26840}+399z^{26842}+1902z^{26844}+360z^{26846}\\&+\,1200z^{26848}+987z^{26850}+504z^{26852}+498z^{26854}+558z^{26856}\\&+\,180z^{26860}+762z^{26862}+63z^{26864}+1884z^{26866}+4257z^{26868}\\&+\,219z^{26870}+18z^{26872}+4131z^{26874}+7587z^{26876}+108z^{26878}\\&+\,372z^{26880}+423z^{26882}+216z^{26884}+348z^{26886}+174z^{26888}\\&+\,435z^{26892}+234z^{26894}+219z^{26896}+1398z^{26898}+2340z^{26900}\\&+\,540z^{26902}+1923z^{26904}+9858z^{26906}+16860z^{26908}+6060z^{26910}\\&+\,3408z^{26912}+8499z^{26914}+12606z^{26916}+9693z^{26918}+8547z^{26920}\\&+\,17154z^{26922}+10989z^{26924}+3456z^{26926}+5004z^{26928}+5808z^{26930}\\&+\,2646z^{26932}+2817z^{26934}+2691z^{26936}+5724z^{26938}+11673z^{26940}\\&+\,4692z^{26942}+4311z^{26944}+7143z^{26946}+6204z^{26948}+3609z^{26950}\\&+\,3528z^{26952}+3798z^{26954}+2313z^{26956}+2823z^{26958}+1134z^{26960}\\&+\,1404z^{26962}+2886z^{26964}+1950z^{26966}+3888z^{26968}+6723z^{26970}\\ \end{aligned}$$
$$\begin{aligned}&+\,9801z^{26972}+8460z^{26974}+12468z^{26976}+29799z^{26978}\\&+\,34686z^{26980}+26046z^{26982}+12531z^{26984}+40473z^{26986}\\&+\,52311z^{26988}+33336z^{26990}+25467z^{26992}+49965z^{26994}\\&+\,42960z^{26996}+24624z^{26998}+35613z^{27000}+50139z^{27002}\\&+\,35523z^{27004}+10953z^{27006}+11805z^{27008}+21216z^{27010}\\&+\,29892z^{27012}+23826z^{27014}+35838z^{27016}+84306z^{27018}\\&+\,90804z^{27020}+58332z^{27022}+52728z^{27024}+90960z^{27026}\\&+\,90555z^{27028}+77877z^{27030}+52029z^{27032}+69576z^{27034}\\&+\,59769z^{27036}+18564z^{27038}+16521z^{27040}+37767z^{27042}\\&+\,42246z^{27044}+35976z^{27046}+45645z^{27048}+62826z^{27050}\\&+\,67551z^{27052}+61803z^{27054}+46197z^{27056}+78516z^{27058}\\&+\,85173z^{27060}+53997z^{27062}+37485z^{27064}+62445z^{27066}\\&+\,66198z^{27068}+36996z^{27070}+42264z^{27072}+75888z^{27074}\\&+\,78588z^{27076}+63609z^{27078}+57777z^{27080}+99816z^{27082}\\&+\,116370z^{27084}+69906z^{27086}+69270z^{27088}+145860z^{27090}\\&+\,182178z^{27092}+137934z^{27094}+127602z^{27096}+299940z^{27098}\\&+\,394902z^{27100}+351789z^{27102}+361329z^{27104}+531045z^{27106} \end{aligned}$$
$$\begin{aligned}&+\,627573z^{27108}+670626z^{27110}+931413z^{27112}+1167318z^{27114}\\&+\,666327z^{27116}+252957z^{27118}+396276z^{27120}+542769z^{27122}\\&+\,432627z^{27124}+289716z^{27126}+195300z^{27128}+255552z^{27130}\\&+\,302040z^{27132}+156429z^{27134}+164088z^{27136}+284487z^{27138}\\&+\,230307z^{27140}+108099z^{27142}+133434z^{27144}+257226z^{27146}\\&+\,244050z^{27148}+194319z^{27150}+104685z^{27152}+178569z^{27154}\\&+\,195930z^{27156}+118290z^{27158}+102033z^{27160}+154602z^{27162}\\&+\,132105z^{27164}+51705z^{27166}+45687z^{27168}+59070z^{27170}\\&+\,53436z^{27172}+46008z^{27174}+44466z^{27176}+106194z^{27178}\\&+\,128205z^{27180}+75735z^{27182}+52125z^{27184}+92301z^{27186}\\&+\,73428z^{27188}+39276z^{27190}+37107z^{27192}+47838z^{27194}\\&+\,29838z^{27196}+7206z^{27198}+11448z^{27200}+16272z^{27202}\\&+\,19356z^{27204}+14925z^{27206}+7203z^{27208}+13557z^{27210}\\&+\,16551z^{27212}+11829z^{27214}+19467z^{27216}+37839z^{27218}\\&+\,35889z^{27220}+15300z^{27222}+8028z^{27224}+8325z^{27226}\\&+\,3975z^{27228}+1485z^{27230}+930z^{27232}+336z^{27234}+855z^{27236}\\&+\,1410z^{27238}+336z^{27240}+180z^{27242}+1821z^{27244}+1128z^{27246}\\&+\,3285z^{27248}+6294z^{27250}+6480z^{27252}+2925z^{27254}+1071z^{27256}\\ \end{aligned}$$
$$\begin{aligned}&+\,720z^{27258}+666z^{27260}+2646z^{27262}+588z^{27264}+2286z^{27266}\\&+\,3357z^{27268}+981z^{27270}+1047z^{27272}+2310z^{27274}+2103z^{27276}\\&+\,1485z^{27278}+2217z^{27280}+2220z^{27282}+459z^{27284}+870z^{27286}\\&+\,228z^{27288}+633z^{27290}+1362z^{27292}+1059z^{27294}+732z^{27296}\\&+\,2268z^{27298}+3399z^{27300}+1518z^{27302}+1770z^{27304}+4122z^{27306}\\&+\,2952z^{27308}+813z^{27310}+1368z^{27312}+2160z^{27314}+1494z^{27316}\\&+\,1167z^{27318}+432z^{27320}+750z^{27322}+1839z^{27324}+747z^{27326}\\&+\,468z^{27328}+957z^{27330}+693z^{27332}+756z^{27334}+582z^{27336}\\&+\,837z^{27338}+324z^{27340}+474z^{27342}+210z^{27344}+420z^{27348}\\&+\,453z^{27350}+252z^{27352}+471z^{27354}+1257z^{27356}+216z^{27358}\\&+\,429z^{27360}+207z^{27362}+42z^{27364}+372z^{27366}+771z^{27368}\\&+\,54z^{27370}+378z^{27372}+675z^{27374}+60z^{27376}+147z^{27378}\\&+\,588z^{27380}+30z^{27382}+183z^{27384}+1281z^{27386}+1980z^{27388}\\&+\,2748z^{27390}+210z^{27392}+2070z^{27394}+5739z^{27396}+96z^{27398}\\&+\,90z^{27400}+312z^{27402}+186z^{27404}+168z^{27406}+468z^{27408}\\&+\,144z^{27410}+411z^{27412}+597z^{27414}+396z^{27416}+2274z^{27418}\\&+\,4596z^{27420}+2688z^{27422}+2292z^{27424}+4842z^{27426}+8445z^{27428}\\&+\,5727z^{27430}+4152z^{27432}+7605z^{27434}+4143z^{27436}+48z^{27438}\\&+\,840z^{27440}+90z^{27444}+1452z^{27446}+1548z^{27448}+1944z^{27450}\\ \end{aligned}$$
$$\begin{aligned}&+\,4857z^{27452}+1962z^{27454}+1413z^{27456}+2910z^{27458}+2112z^{27460}\\&+\,846z^{27462}+1134z^{27464}+1680z^{27466}+327z^{27468}+1113z^{27470}\\&+\,315z^{27472}+3z^{27474}+813z^{27476}+828z^{27478}+399z^{27480}\\&+\,837z^{27482}+1257z^{27484}+651z^{27486}+531z^{27488}+807z^{27490}\\&+\,237z^{27492}+429z^{27494}+606z^{27496}+633z^{27498}+162z^{27500}\\&+\,558z^{27502}+819z^{27504}+582z^{27506}+630z^{27508}+2550z^{27510}\\&+\,2133z^{27512}+2718z^{27514}+2913z^{27516}+1233z^{27518}+1377z^{27520}\\&+\,3420z^{27522}+1854z^{27524}+1035z^{27526}+987z^{27528}+1431z^{27530}\\&+\,180z^{27532}+12z^{27534}+507z^{27536}+288z^{27538}+648z^{27540}\\&+\,1365z^{27542}+1548z^{27544}+2736z^{27546}+4416z^{27548}+3786z^{27550}\\&+\,3852z^{27552}+5406z^{27554}+7068z^{27556}+5040z^{27558}+4860z^{27560}\\&+\,4962z^{27562}+792z^{27564}+60z^{27566}+540z^{27568}+144z^{27570}\\&+\,414z^{27572}+825z^{27574}+288z^{27576}+990z^{27578}+1575z^{27580}\\&+\,1347z^{27582}+1950z^{27584}+3444z^{27586}+3648z^{27588}+1584z^{27590}\\&+\,1266z^{27592}+1887z^{27594}+504z^{27596}+1197z^{27598}+186z^{27600}\\&+\,492z^{27604}+222z^{27606}+114z^{27610}+192z^{27612}+321z^{27614}\\&+\,903z^{27616}+1143z^{27618}+1137z^{27620}+642z^{27622}+252z^{27624}\\&+\,243z^{27626}+216z^{27628}+444z^{27630}+18z^{27632}+84z^{27634}\\ \end{aligned}$$
$$\begin{aligned}&\,\,+\,237z^{27636}+18z^{27638}+18z^{27640}+57z^{27642}+18z^{27644}+123z^{27648}\\&\,\,+\,24z^{27650}+123z^{27654}+12z^{27656}+90z^{27660}+42z^{27666}+9z^{27672}\\&\,\,+\,30z^{27710}+15z^{27716}+288z^{27718}+288z^{27720}+864z^{27722}\\&\,\,+\,144z^{27724}+576z^{27726}+36z^{27742}+9z^{27744}+42z^{27748}+234z^{27750}\\&\,\,+\,144z^{27752}+438z^{27754}+183z^{27756}+288z^{27758}+87z^{27762}\\&\,\,+\,69z^{27768}+198z^{27774}+165z^{27780}+69z^{27786}+15z^{27792}+27z^{27798}\\&\,\,+\,138z^{27804}+36z^{27806}+54z^{27810}+42z^{27812}+66z^{27816}+6z^{27818}\\&\,\,+\,177z^{27822}+240z^{27828}+132z^{27834}+120z^{27838}+195z^{27840}\\&\,\,+\,60z^{27844}+264z^{27846}+261z^{27852}+99z^{27858}+156z^{27864}\\&\,\,+\,351z^{27870}+231z^{27876}+84z^{27882}+36z^{27884}+33z^{27888}\\&\,\,+\,15z^{27894}+108z^{27902}+42z^{27908}+54z^{27914}+72z^{27916}+60z^{27920}\\&\,\,+\,30z^{27926}+120z^{27966}+60z^{27972}+60z^{27998}+30z^{28004}+72z^{28006}\\&\,\,+\,144z^{28008}+252z^{28010}+36z^{28012}+144z^{28038}+288z^{28040}\\&\,\,+\,504z^{28042}+72z^{28044}+9z^{28098}+21z^{28224}+39z^{28230}+39z^{28236}\\&\,\,+\,24z^{28242}+12z^{28248}+18z^{28254}+21z^{28260}+3z^{28266}+3z^{28278}\\&\,\,+\,21z^{28284}+36z^{28286}+30z^{28290}+42z^{28292}+27z^{28296}+6z^{28298}\\&\,\,+\,45z^{28302}+57z^{28308}+54z^{28314}+36z^{28320}+57z^{28326}+48z^{28332}\\&\,\,+\,12z^{28338}+15z^{28344}+18z^{28350}+27z^{28356}+33z^{28368}+27z^{28374}\\&\,\,+\,12z^{28380}+18z^{28386}+3z^{31848}+3z^{32520}+12z^{33120}+6z^{33126} \end{aligned}$$
$$\begin{aligned}&+\,39z^{33132}+18z^{33138}+12z^{33144}+3z^{33510}+45z^{33878}+3z^{34014}\\&+\,15z^{34422},\\ Wt^{\perp }_{36150}(y)= & {} 1+553592502981y^4+6606240021787176y^5+\cdots . \end{aligned}$$

Appendix E

The [15816, 11, 10992]-cap code given in Sect. 6.4 has weight polynomial

$$\begin{aligned} Wt_{15816}(z)= & {} 1+72z^{10992}+63z^{10998}+36z^{11004}+36z^{11016}+171z^{11028}+27z^{11034}\\&+\,180z^{11040}+72z^{11046}+180z^{11048}+72z^{11052}+108z^{11056}\\&+\,432z^{11060}+297z^{11062}+432z^{11064}+108z^{11068}+252z^{11088}\\&+\,216z^{11094}+540z^{11096}+72z^{11100}+1296z^{11124}+324z^{11126}\\&+\,54z^{11136}+84z^{11184}+9z^{11190}+12z^{11208}+45z^{11220}+9z^{11226}\\&+\,108z^{11248}+27z^{11254}+144z^{11256}+12z^{11376}+387z^{11604}\\&+\,135z^{11610}+540z^{11616}+432z^{11620}+36z^{11652}+216z^{11708}\\&+\,648z^{11714}+2160z^{11716}+648z^{11720}+2592z^{11744}+12636z^{11746}\\&+\,27216z^{11748}+1512z^{11756}+2592z^{11760}+6156z^{11762}+15552z^{11764}\\&+\,1512z^{11768}+2808z^{11772}+1944z^{11778}+2160z^{11780}+1944z^{11784}\\&+\,69z^{11796}+2160z^{11800}+45z^{11802}+90720z^{11804}+116640z^{11806}\\&+\,125208z^{11808}+37908z^{11810}+48096z^{11812}+11880z^{11814}\\&+\,11124z^{11816}+97632z^{11820}+77760z^{11822}+57456z^{11824}\\&+\,28836z^{11826}+38016z^{11828}+15912z^{11832}+12960z^{11836}\\&+\,12960z^{11838}+4320z^{11840}+6060z^{11844}+6480z^{11848}+5040z^{11850}\\&+\,163260z^{11856}+347328z^{11858}+414729z^{11860}+86328z^{11862}\\&+\,46188z^{11864}+405z^{11866}+330696z^{11868}+349920z^{11870}\\&+\,358668z^{11872}+146880z^{11874}+203472z^{11876}+23112z^{11878}\\&+\,37116z^{11880}+209736z^{11884}+233280z^{11886}+129600z^{11888}\\&+\,576z^{11892}+38880z^{11900}+38880z^{11902}+13005z^{11904}+108z^{11908}\\&+\,2160z^{11912}+3780z^{11920}+3240z^{11926}+5940z^{11928}+1080z^{11932}\\&+\,72z^{11964}+135z^{11968}+216z^{11970}+720z^{11972}+216z^{11976}\\&+\,3672z^{12000}+4212z^{12002}+9072z^{12004}+648z^{12006}+2268z^{12008}\\&+\,720z^{12012}+1080z^{12016}+2052z^{12018}+5184z^{12020}+189z^{12022}\\&+\,648z^{12024}+1044z^{12028}+648z^{12034}+1776z^{12036}+756z^{12040}\\&+\,336z^{12042}+1092z^{12048}+207z^{12052}+648z^{12054}+1620z^{12056}\\&+\,135z^{12058}+216z^{12060}+9072z^{12064}+12636z^{12066}+14688z^{12068}\\&+\,3672z^{12076}+5076z^{12080}+6156z^{12082}+8208z^{12084}+891z^{12086}\\&+\,2808z^{12088}+324z^{12092}+18z^{12096}+36z^{12100}+3888z^{12148} \end{aligned}$$
$$\begin{aligned}&+\,972z^{12150}+252z^{12208}+27z^{12214}+36z^{12232}+324z^{12272}\\&+\,81z^{12278}+432z^{12280}+36z^{12400}+3z^{14400}+57z^{14868}+9z^{14874}\\&+\,36z^{14880}+15z^{15060}+3z^{15066}. \end{aligned}$$

The weight polynomial of its dual is

$$\begin{aligned} Wt^{\perp }_{15816}(y)=1+82032878250y^{4}+\cdots . \end{aligned}$$

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Li, R., Fu, Q., Guo, L. et al. Construction of quantum caps in projective space PG(r, 4) and quantum codes of distance 4. Quantum Inf Process 15, 689–720 (2016). https://doi.org/10.1007/s11128-015-1204-9

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