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Ternary optimal quantum codes constructed from caps in \(PG(k,9)(k \ge 2)\)

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Abstract

By recursive construction and computer-supported search method, the spectrum of quantum caps in projective space \(PG(k,9)(k \ge 2)\) is determined. Through Hermitian construction, the obtained quantum caps are used to construct ternary quantum codes of \(d=4\). As a result, for each integer n satisfying \(n\ge 10\), a quantum n-cap in \(PG(k-1,9)\) with suitable k and its related ternary \([[n,n-2k,4]]\) quantum code are constructively proven to exist. According to quantum GV bound and quantum Hamming bound, all quantum codes are optimal. In addition, while constructing quantum caps, the cardinalities of maximal caps in PG(7, 9) and PG(8, 9) are also improved.

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Acknowledgements

This work is supported by the Graduate Scientific Research Foundation of Department of Basic Sciences of Air Force Engineering University and National Natural Science Foundation of China (Nos.11801564, 11901579, U21A20428).

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  1. Department of Basic Sciences, Air Force Engineering University, Xi’an, 710051, People’s Republic of China

    Husheng Li, Ruihu Li, Qiang Fu & Xiuzhen Zhan

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  1. Husheng Li
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  2. Ruihu Li
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Appendix

Appendix

The equivalent 82-cap \(G_{4,82}'\) mentioned in Sect. 3.2 is as follows:

$$\begin{aligned} G_{4,82}'= & {} (S_{1}',S_{2}'), \\ S_{1}'= & {} \left( {\begin{array}{*{20}{c}} {111111111111111111111111111111111111111111}\\ {085236662404010162228653083426814237154432}\\ {064406328514063306844806815270606583151145}\\ {067303510672164071464274256265750754124143} \end{array}} \right) , \\ S_{2}'= & {} \left( {\begin{array}{*{20}{c}} {1111111111111111111111111111111111111110}\\ {1100365855803553528004468115438614326184}\\ {2113135348424221832282458060651825065338}\\ {0340510535712537013274626132173650243681} \end{array}} \right) . \end{aligned}$$

The 212-cap \(G_{5,212}'\) mentioned in Sect. 3.2 is as follows:

$$\begin{aligned}&{{G}_{5,212}}\mathrm{{ = }}\left( {\begin{array}{*{20}{c}} {{{G}_{5,198}}} \end{array}\;\left| {\begin{array}{*{20}{c}} 1&{} \cdots &{}1&{}0&{}0&{}0\\ {{\beta _{199}}}&{} \cdots &{}{{\beta _{209}}}&{}{{\beta _{210}}}&{}{{\beta _{211}}}&{}{{\beta _{212}}} \end{array}} \right. } \right) ,{G_{5,198}} = \left( {{Q_{65}},{Q_{66}},{Q_{67}}} \right) ,\\&\left( {\begin{array}{*{20}{c}} 1&{} \cdots &{}1&{}0&{}0&{}0\\ {{\beta _{199}}}&{} \cdots &{}{{\beta _{209}}}&{}{{\beta _{210}}}&{}{{\beta _{211}}}&{}{{\beta _{212}}} \end{array}} \right) = \left( \begin{array}{l} \mathrm{{11111111111000}}\\ \mathrm{{51426388220420}}\\ \mathrm{{26534870806030}}\\ \mathrm{{51426382545120}}\\ \mathrm{{06357756321021}} \end{array} \right) , \\&{Q_{65}=\left( \begin{array}{cccccc} 11111111111111111111111111111111111111111111111111111111111111111\\ 02516018305827800514601830582780120752477126614326385247712661430\\ 06540853721626381073406723516413802772160853051445682351406770821\\ 01400342413673480514034241367348084785718207560226388571820756027\\ 04600057102030600063005710203060000036638448577557753663844857757\\ \end{array} \right) }, \\&{Q_{66}=\left( \begin{array}{cccccc} 111111111111111111111111111111111111111111111111111111111111111111\\ 847836270153610047630244015861051237185236854775123718523685427743\\ 727452187466625627115878305170307460861132547538305320462715864158\\ 736757080156043725485464015204308610317236887710861031723688751635\\ 063007501520438754884637152643854784683215784065478468321578436360\\ \end{array} \right) }, \\&{Q_{67}=\left( \begin{array}{cccccc} 1111111111111111111111111111111111111111111111111111111111111111111\\ 4763024413461250606258305027852137815521378156074432060744330478306\\ 3257254046357320461085376142872548546158774380861031732042531841237\\ 2548546422056142828146413873406810137068101374752684547526864367528\\ 5488463702100557777553366448800750084007500846336122163361200483683\\ \end{array} \right) }. \end{aligned}$$

The 840-cap \(G_{6,840}\) mentioned in Sect. 3.3 is as follows:

$$\begin{aligned} {{G}_{6,840}}= & {} \left( {\begin{array}{*{20}{c}} 1&{} \cdots &{}1&{}0&{} \cdots &{}0\\ {{\gamma _1}}&{} \cdots &{}{{\gamma _{830}}}&{}{{\gamma _{831}}}&{} \cdots &{}{{\gamma _{840}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{1_{70}}}&{}{{1_{70}}}&{} \cdots &{}{{1_{70}}}&{}{{1_{60}}}&{}{{0_{10}}}\\ {{\mathbf{{Y}}_1}}&{}{{\mathbf{{Y}}_2}}&{} \cdots &{}{{\mathbf{{Y}}_{11}}}&{}{{\mathbf{{Y}}_{12}}}&{}\mathbf{{Z}} \end{array}} \right) ,\\ \mathbf{{Y}}_1= & {} \left( \begin{array}{cccccc} 3186556813186556813386556813316556813318556813318656813318656813318655\\ 0474307535014230234505225084480324105432034740535705767085860168702816\\ 1755142788538311665861138358562415571887848132162481544257722337476462\\ 0474307535474307535074307535044307535047307535047407535047437535047430\\ 1755142788755142788155142788175142788175142788175542788175512788175514\\ \end{array} \right) , \\ \mathbf{{Y}}_2= & {} \left( \begin{array}{cccccc} 8133186556133186556833186556811425787524425787524125787524145787524142\\ 0488407117078610618207675068587336168187672374462358121857474473276326\\ 6747332264524451827712318484263035770118702781068260187207281077530381\\ 5350474307350474307550474307537336168187336168187736168187736168187733\\ 7881755142881755142781755142783035770118035770118335770118305770118303\\ \end{array} \right) , \\ \mathbf{{Y}}_3= & {} \left( \begin{array}{cccccc} 7875241425875241425775241425785241425787241425787541425787521164755746\\ 8616337781342455663288316351654758574212536138856165542432366088510146\\ 2015250717502246067520341501541051430245606422055770525102712854253384\\ 1681877336681877336181877336161877336168877336168177336168186088510146\\ 7701183035701183035701183035771183035770183035770183035770112854253384\\ \end{array} \right) , \\ \mathbf{{Y}}_4= & {} \left( \begin{array}{cccccc} 1647557461647557461147557461167557461164557461164757461164757461164755\\ 6075710561402772041150175706161015880664202833044720732305751082280155\\ 2727156136316517272683352458245637764835766481478173263151525151362372\\ 0885101466885101466085101466085101466088101466088501466088511466088510\\ 8542533842542533842842533842852533842854533842854233842854253842854253\\ \end{array} \right) , \\ \mathbf{{Y}}_5= & {} \left( \begin{array}{cccccc} 4611647557611647557463747365853747365856747365856347365856377365856374\\ 5032370257403382027471345713455417654176333333333314567145673175431754\\ 8741846671384677365574210742105273052730372503725012470124703486034860\\ 4660885101660885101471345713451345713457345713457145713457135713457134\\ 8428542533428542533874210742104210742107210742107410742107420742107421\\ \end{array} \right) , \end{aligned}$$
$$\begin{aligned} \mathbf{{Y}}_6= & {} \left( \begin{array}{cccccc} 3658563747658563747358563747368563747365563747365883387211273387211278\\ 5268752687782537825366666666662873528735625786257866088510141607571056\\ 5812058120715607156065170651702185021850684306843042854253386272715613\\ 7134571345134571345734571345714571345713571345713466088510146088510146\\ 7421074210421074210721074210741074210742074210742142854253382854253384\\ \end{array} \right) , \\ \mathbf{{Y}}_7= & {} \left( \begin{array}{cccccc} 3872112783872112783372112783382112783387112783387212783387212783387211\\ 1402772041650175706141015880667202833044520732305751082280157503237025\\ 6316517272483352458255637764831766481478273263151525151362371874184667\\ 0885101466885101466085101466085101466088101466088501466088511466088510\\ 8542533842542533842842533842852533842854533842854233842854253842854253\\ \end{array} \right) , \\ \mathbf{{Y}}_8= & {} \left( \begin{array}{cccccc} 7833872112824283616324283616384283616382283616382483616382423616382428\\ 4403382027877336168123672374464758121857264473276381861633773234245566\\ 5384677365183035770182702781062860187207811077530317201525077550224606\\ 4660885101877336168177336168187336168187336168187736168187736168187733\\ 8428542533183035770183035770113035770118035770118335770118305770118303\\ \end{array} \right) , \\ \mathbf{{Y}}_9= & {} \left( \begin{array}{cccccc} 6163824283163824283663824283613824283616372273144172273144132273144137\\ 6588316351124758574261536138853665542432535047430734501423024480522508\\ 5420341501451051430257606422057170525102788175514265853831168566113835\\ 1681877336681877336181877336161877336168535047430735047430755047430753\\ 7701183035701183035701183035771183035770788175514288175514278175514278\\ \end{array} \right) , \\ \mathbf{{Y}}_{10}= & {} \left( \begin{array}{cccccc} 2731441372731441372231441372271441372273441372273141372273141372273144\\ 4320324105357034740558605767088160168702117048840718207861068580767506\\ 8872415571624848132177281544254622337476264674733227752445184261231848\\ 0474307535474307535074307535044307535047307535047407535047437535047430\\ 1755142788755142788155142788175142788175142788175542788175512788175514\\ \end{array} \right) , \\ \mathbf{{Y}}_{11}= & {} \left( \begin{array}{cccccc} 5258642468258642468558642468528642468525642468525842468525862468525864\\ 5864485265232355158151782871568510532321621844685547137743887274216167\\ 4204232063550337808448086330553603324024145826705361036810815404877066\\ 5864480265864480265564480265584480265586480265586480265586440265586448\\ 4204230063204230063404230063424230063420230063420430063420420063420423\\ \end{array} \right) , \\ \mathbf{{Y}}_{12}= & {} \left( \begin{array}{cccccc} 468525864268525864248525864246000000006078403873540000000000\\ 452737254161612472078347731248000000000000000000001025520177\\ 660778404318018630663507628741000000000000000000005367276351\\ 265586448065586448025586448026187763816336163187780000000000\\ 063420423063420423003420423006118353077035770118300000000000\\ \end{array} \right) , \\ \mathbf{{Z}}= & {} \left( \begin{array}{cccccc} 2222222222\\ 8262841314\\ 0255201771\\ 4444444444\\ 1111111111\\ \end{array} \right) . \end{aligned}$$

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Li, H., Li, R., Fu, Q. et al. Ternary optimal quantum codes constructed from caps in \(PG(k,9)(k \ge 2)\). Quantum Inf Process 21, 96 (2022). https://doi.org/10.1007/s11128-022-03437-5

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