# American Institute of Mathematical Sciences

November  2020, 14(4): 613-630. doi: 10.3934/amc.2020034

## New and updated semidefinite programming bounds for subspace codes

 1 Department of Communications and Networking, Aalto University, Finland 2 Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Belgium

* Corresponding author: Daniel Heinlein

Received  September 2018 Revised  July 2019 Published  November 2019

Fund Project: The first author is supported by the Academy of Finland, Grant #289002. The second author is supported by a postdoctoral fellowship of the Research Foundation - Flanders (FWO)

We show that $A_2(7, 4) \leq 388$ and, more generally, $A_q(7, 4) \leq (q^2-q+1) [7] + q^4 - 2q^3 + 3q^2 - 4q + 4$ by semidefinite programming for $q \leq 101$. Furthermore, we extend results by Bachoc et al. on SDP bounds for $A_2(n, d)$, where $d$ is odd and $n$ is small, to $A_q(n, d)$ for small $q$ and small $n$.

Citation: Daniel Heinlein, Ferdinand Ihringer. New and updated semidefinite programming bounds for subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 613-630. doi: 10.3934/amc.2020034
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Here $\varphi = q^2+1$ and $\psi = q^2-q+1$
 $A_{abc}$ $m_{abc}/|X_a|$ $\Delta_0(A_{abc})$ $\Delta_1(A_{abc})$ $\Delta_2(A_{abc})$ $\Delta_3(A_{abc})$ $A_{110}$ $1$ $E_{11}$ $E_{11}$ $A_{111}$ $q[6]$ $q[6] E_{11}$ $-E_{11}$ $A_{120}$ $[6]$ $[2] \sqrt{ \psi[3] } E_{12}$ $\sqrt{q [5] } E_{12}$ $A_{121}$ $q^2\psi [3] [5]$ $q^2 [5] \sqrt{ \psi[3] } E_{12}$ $-\sqrt{q [5] } E_{12}$ $A_{130}$ $\left[ \begin{array}{l} {6}\\{2} \end{array} \right]$ $[3] \sqrt{ \psi [5] } E_{13}$ $q \sqrt{ \varphi [5]} E_{13}$ $A_{131}$ $q^3(q^3+1)\left[ \begin{array}{l}{5}\\{2}\end{array} \right]$ $q^3 [4] \sqrt{ \psi [5] } E_{13}$ $-q \sqrt{ \varphi [5]}E_{13}$ $A_{140}$ $\left[ \begin{array}{l} 6\\3\end{array} \right]$ $[4] \sqrt{ \psi [5] } E_{14}$ $q \sqrt{q \varphi [5]} E_{14}$ $A_{141}$ $q^4 \psi [3] [5]$ $q^4 [3] \sqrt{ \psi [5] } E_{14}$ $-q \sqrt{q \varphi [5]} E_{14}$ $A_{150}$ $\left[ \begin{array}{l} {6}\\{4} \end{array} \right]$ $[5] \sqrt{ \psi [3] } E_{15}$ $q^2 \sqrt{ [5] } E_{15}$ $A_{151}$ $q^5[6]$ $q^5 [2] \sqrt{ \psi [3] } E_{15}$ $-q^2 \sqrt{ [5] } E_{15}$ $A_{160}$ $\left[ \begin{array}{l} 6\\5\end{array} \right]$ $[6] E_{16}$ $q^{5/2} E_{16}$ $A_{161}$ $q^6$ $q^6 E_{16}$ $-q^{5/2} E_{16}$ $A_{220}$ $1$ $E_{22}$ $E_{22}$ $E_{22}$ $A_{221}$ $q[2] [5]$ $q[2] [5] E_{22}$ $(q^2 [4] - 1)E_{22}$ $-[2]E_{22}$ $A_{222}$ $q^4\varphi [5]$ $q^4\varphi [5] E_{22}$ $-q^2 [4] E_{22}$ $qE_{22}$ $A_{230}$ $[5]$ $\sqrt{ [3] [5] } E_{23}$ $[2] \sqrt{q\varphi} E_{23}$ $q \sqrt{ [3] } E_{23}$ $A_{231}$ $q^2 [4][5]$ $q^2 [4] \sqrt{ [3] [5] } E_{23}$ $(q^3[3] - [2]) \sqrt{q\varphi} E_{23}$ $-q[2] \sqrt{ [3] } E_{23}$ $A_{232}$ $q^6\varphi[5]$ $q^6 \varphi \sqrt{ [3] [5] } E_{23}$ $-q^3[3] \sqrt{q\varphi} E_{23}$ $q^2\sqrt{ [3] } E_{23}$ $A_{240}$ $\varphi[5]$ $\varphi \sqrt{ [3] [5] } E_{24}$ $q [3] \sqrt{\varphi} E_{24}$ $q^2 \sqrt{[3]}E_{24}$ $A_{241}$ $q^3 [4][5]$ $q^3 [4] \sqrt{ [3] [5] } E_{24}$ $q (q^4[2]-[3]) \sqrt{\varphi} E_{24}$ $-q^2[2] \sqrt{[3]} E_{24}$ $A_{242}$ $q^8 [5]$ $q^8 \sqrt{ [3] [5] } E_{24}$ $-q^5 [2] \sqrt{\varphi} E_{24}$ $q^3 \sqrt{[3]} E_{24}$ $A_{250}$ $\varphi [5]$ $\varphi [5] E_{25}$ $q^{3/2} [4] E_{25}$ $q^3 E_{25}$ $A_{251}$ $q^4 [2] [5]$ $q^4 [2] [5] E_{25}$ $q^{3/2}(q^5 - [4]) E_{25}$ $-[2] q^3E_{25}$ $A_{252}$ $q^{10}$ $q^{10} E_{25}$ $-q^{13/2} E_{25}$ $q^4 E_{25}$ $A_{330}$ $1$ $E_{33}$ $E_{33}$ $E_{33}$ $E_{33}$ $A_{331}$ $q [3] [4]$ $q [3] [4] E_{33}$ $(q^2[2][3] - 1)E_{33}$ $(q^2-1) [3] E_{33}$ $-[3] E_{33}$ $A_{332}$ $q^4 \varphi [3]^2$ $q^4 \varphi [3]^2 E_{33}$ $q^2 [3] (q^4-q-1)E_{33}$ $-q[3] (q^2+q-1)E_{33}$ $q [3] E_{33}$ $A_{333}$ $q^9[4]$ $q^9[4] E_{33}$ $-q^6 [3] E_{33}$ $q^4 [2]E_{33}$ $-q^3 E_{33}$ $A_{340}$ $[4]$ $[4] E_{34}$ $[3] \sqrt{q} E_{34}$ $q [2] E_{34}$ $\sqrt{q^3 } E_{34}$ $A_{341}$ $q^2\varphi[3]^2$ $q^2 \varphi [3]^2 E_{34}$ $[3] (q^3[2]-1) \sqrt{q} E_{34}$ $q [3](q^2-q-1) E_{34}$ $-[3] \sqrt{q^3 } E_{34}$ $A_{342}$ $q^6[3] [4]$ $q^6 [3] [4] E_{34}$ $q^3(q^5-[2][3]) \sqrt{q} E_{34}$ $-q^2 (q^3-1) [2] E_{34}$ $q[3] \sqrt{q^3 }E_{34}$ $A_{343}$ $q^{12}$ $q^{12} E_{34}$ $-q^8 \sqrt{q} E_{34}$ $q^6 E_{34}$ $-q^3\sqrt{q^3 }E_{34}$ $f_s$ $1$ $[7]-1$ $\left[ \begin{array}{l} 7\\2\end{array} \right] - [7]$ $\left[ \begin{array}{l} 7\\3\end{array} \right] - \left[ \begin{array}{l} 7\\2\end{array} \right]$
 $A_{abc}$ $m_{abc}/|X_a|$ $\Delta_0(A_{abc})$ $\Delta_1(A_{abc})$ $\Delta_2(A_{abc})$ $\Delta_3(A_{abc})$ $A_{110}$ $1$ $E_{11}$ $E_{11}$ $A_{111}$ $q[6]$ $q[6] E_{11}$ $-E_{11}$ $A_{120}$ $[6]$ $[2] \sqrt{ \psi[3] } E_{12}$ $\sqrt{q [5] } E_{12}$ $A_{121}$ $q^2\psi [3] [5]$ $q^2 [5] \sqrt{ \psi[3] } E_{12}$ $-\sqrt{q [5] } E_{12}$ $A_{130}$ $\left[ \begin{array}{l} {6}\\{2} \end{array} \right]$ $[3] \sqrt{ \psi [5] } E_{13}$ $q \sqrt{ \varphi [5]} E_{13}$ $A_{131}$ $q^3(q^3+1)\left[ \begin{array}{l}{5}\\{2}\end{array} \right]$ $q^3 [4] \sqrt{ \psi [5] } E_{13}$ $-q \sqrt{ \varphi [5]}E_{13}$ $A_{140}$ $\left[ \begin{array}{l} 6\\3\end{array} \right]$ $[4] \sqrt{ \psi [5] } E_{14}$ $q \sqrt{q \varphi [5]} E_{14}$ $A_{141}$ $q^4 \psi [3] [5]$ $q^4 [3] \sqrt{ \psi [5] } E_{14}$ $-q \sqrt{q \varphi [5]} E_{14}$ $A_{150}$ $\left[ \begin{array}{l} {6}\\{4} \end{array} \right]$ $[5] \sqrt{ \psi [3] } E_{15}$ $q^2 \sqrt{ [5] } E_{15}$ $A_{151}$ $q^5[6]$ $q^5 [2] \sqrt{ \psi [3] } E_{15}$ $-q^2 \sqrt{ [5] } E_{15}$ $A_{160}$ $\left[ \begin{array}{l} 6\\5\end{array} \right]$ $[6] E_{16}$ $q^{5/2} E_{16}$ $A_{161}$ $q^6$ $q^6 E_{16}$ $-q^{5/2} E_{16}$ $A_{220}$ $1$ $E_{22}$ $E_{22}$ $E_{22}$ $A_{221}$ $q[2] [5]$ $q[2] [5] E_{22}$ $(q^2 [4] - 1)E_{22}$ $-[2]E_{22}$ $A_{222}$ $q^4\varphi [5]$ $q^4\varphi [5] E_{22}$ $-q^2 [4] E_{22}$ $qE_{22}$ $A_{230}$ $[5]$ $\sqrt{ [3] [5] } E_{23}$ $[2] \sqrt{q\varphi} E_{23}$ $q \sqrt{ [3] } E_{23}$ $A_{231}$ $q^2 [4][5]$ $q^2 [4] \sqrt{ [3] [5] } E_{23}$ $(q^3[3] - [2]) \sqrt{q\varphi} E_{23}$ $-q[2] \sqrt{ [3] } E_{23}$ $A_{232}$ $q^6\varphi[5]$ $q^6 \varphi \sqrt{ [3] [5] } E_{23}$ $-q^3[3] \sqrt{q\varphi} E_{23}$ $q^2\sqrt{ [3] } E_{23}$ $A_{240}$ $\varphi[5]$ $\varphi \sqrt{ [3] [5] } E_{24}$ $q [3] \sqrt{\varphi} E_{24}$ $q^2 \sqrt{[3]}E_{24}$ $A_{241}$ $q^3 [4][5]$ $q^3 [4] \sqrt{ [3] [5] } E_{24}$ $q (q^4[2]-[3]) \sqrt{\varphi} E_{24}$ $-q^2[2] \sqrt{[3]} E_{24}$ $A_{242}$ $q^8 [5]$ $q^8 \sqrt{ [3] [5] } E_{24}$ $-q^5 [2] \sqrt{\varphi} E_{24}$ $q^3 \sqrt{[3]} E_{24}$ $A_{250}$ $\varphi [5]$ $\varphi [5] E_{25}$ $q^{3/2} [4] E_{25}$ $q^3 E_{25}$ $A_{251}$ $q^4 [2] [5]$ $q^4 [2] [5] E_{25}$ $q^{3/2}(q^5 - [4]) E_{25}$ $-[2] q^3E_{25}$ $A_{252}$ $q^{10}$ $q^{10} E_{25}$ $-q^{13/2} E_{25}$ $q^4 E_{25}$ $A_{330}$ $1$ $E_{33}$ $E_{33}$ $E_{33}$ $E_{33}$ $A_{331}$ $q [3] [4]$ $q [3] [4] E_{33}$ $(q^2[2][3] - 1)E_{33}$ $(q^2-1) [3] E_{33}$ $-[3] E_{33}$ $A_{332}$ $q^4 \varphi [3]^2$ $q^4 \varphi [3]^2 E_{33}$ $q^2 [3] (q^4-q-1)E_{33}$ $-q[3] (q^2+q-1)E_{33}$ $q [3] E_{33}$ $A_{333}$ $q^9[4]$ $q^9[4] E_{33}$ $-q^6 [3] E_{33}$ $q^4 [2]E_{33}$ $-q^3 E_{33}$ $A_{340}$ $[4]$ $[4] E_{34}$ $[3] \sqrt{q} E_{34}$ $q [2] E_{34}$ $\sqrt{q^3 } E_{34}$ $A_{341}$ $q^2\varphi[3]^2$ $q^2 \varphi [3]^2 E_{34}$ $[3] (q^3[2]-1) \sqrt{q} E_{34}$ $q [3](q^2-q-1) E_{34}$ $-[3] \sqrt{q^3 } E_{34}$ $A_{342}$ $q^6[3] [4]$ $q^6 [3] [4] E_{34}$ $q^3(q^5-[2][3]) \sqrt{q} E_{34}$ $-q^2 (q^3-1) [2] E_{34}$ $q[3] \sqrt{q^3 }E_{34}$ $A_{343}$ $q^{12}$ $q^{12} E_{34}$ $-q^8 \sqrt{q} E_{34}$ $q^6 E_{34}$ $-q^3\sqrt{q^3 }E_{34}$ $f_s$ $1$ $[7]-1$ $\left[ \begin{array}{l} 7\\2\end{array} \right] - [7]$ $\left[ \begin{array}{l} 7\\3\end{array} \right] - \left[ \begin{array}{l} 7\\2\end{array} \right]$
SDP bounds on $A_2(n, d)$
 $d \setminus n$ 8 9 10 11 12 13 14 3 9191 107419 2531873 57201557 2685948795 119527379616 11215665059647 4 6479 53710 1705394 28600778 1816165540 59763689822 7496516673358 5 327 2458 48255 660265 26309023 688127334 54724534275 6 260 1240 38455 330133 21362773 344063682 43890879895 7 1219 8844 314104 4678401 330331546 8 1090 4480 279476 2343888 292988615 9 4483 34058 2298622 10 4226 17133 2164452 11 259 17155 12 16642
 $d \setminus n$ 8 9 10 11 12 13 14 3 9191 107419 2531873 57201557 2685948795 119527379616 11215665059647 4 6479 53710 1705394 28600778 1816165540 59763689822 7496516673358 5 327 2458 48255 660265 26309023 688127334 54724534275 6 260 1240 38455 330133 21362773 344063682 43890879895 7 1219 8844 314104 4678401 330331546 8 1090 4480 279476 2343888 292988615 9 4483 34058 2298622 10 4226 17133 2164452 11 259 17155 12 16642
SDP bounds on $A_3(n, d)$
 $d \setminus n$ 6 7 8 9 10 11 12 3 967 15394 760254 34143770 5026344026 675225312722 298950313257852 4 788 7696 627384 17071886 4112061519 337612656529 244829520433920 5 166 7222 123535 16008007 818518696 320387589445 6 6727 61962 14893814 409259348 298571221318 7 490 61002 1076052 400831735 8 59539 539351 391178436 9 1462 537278 10 532903
 $d \setminus n$ 6 7 8 9 10 11 12 3 967 15394 760254 34143770 5026344026 675225312722 298950313257852 4 788 7696 627384 17071886 4112061519 337612656529 244829520433920 5 166 7222 123535 16008007 818518696 320387589445 6 6727 61962 14893814 409259348 298571221318 7 490 61002 1076052 400831735 8 59539 539351 391178436 9 1462 537278 10 532903
SDP bounds on $A_4(n, d)$
 $d \setminus n$ 6 7 8 9 10 11 3 4772 142313 20482322 2341621613 1343547758223 614496020025690 4 4231 71156 18245203 1170810807 1194101275238 307248010015067 5 516 68117 2132181 1122729102 140323867490 6 66054 1067796 1088550221 70161933745 7 2052 1058831 33669242 8 1050630 16847095 9 8196
 $d \setminus n$ 6 7 8 9 10 11 3 4772 142313 20482322 2341621613 1343547758223 614496020025690 4 4231 71156 18245203 1170810807 1194101275238 307248010015067 5 516 68117 2132181 1122729102 140323867490 6 66054 1067796 1088550221 70161933745 7 2052 1058831 33669242 8 1050630 16847095 9 8196
SDP bounds on $A_5(n, d)$
 $d \setminus n$ 6 7 8 9 10 3 17179 821170 277100135 64262978412 108238287449582 4 15883 410585 256754528 32131489207 100215014898311 5 1254 398154 19675409 31196584033 6 391883 9847885 30703887393 7 6254 9803150 8 9771883
 $d \setminus n$ 6 7 8 9 10 3 17179 821170 277100135 64262978412 108238287449582 4 15883 410585 256754528 32131489207 100215014898311 5 1254 398154 19675409 31196584033 6 391883 9847885 30703887393 7 6254 9803150 8 9771883
SDP bounds on $A_7(n, d)$
 $d \setminus n$ 6 7 8 9 10 3 123239 11807778 14753449680 9728400942608 85039309360944189 4 118347 5903889 14176726504 4864200471305 81703574152063079 5 4806 5803270 566262547 4784663914039 6 5769615 283240686 4756893963688 7 33618 282744208 8 282508875
 $d \setminus n$ 6 7 8 9 10 3 123239 11807778 14753449680 9728400942608 85039309360944189 4 118347 5903889 14176726504 4864200471305 81703574152063079 5 4806 5803270 566262547 4784663914039 6 5769615 283240686 4756893963688 7 33618 282744208 8 282508875
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