# Details for constraints

### upper bound, not recursive

HKK_theorem_3_3_i_upper_bound_A_q_6_4
$$A_q(6,4)\le(q^3+1)^2$$ for all $$q\ge 3$$.
HonoldKiermaierKurz20163 (Theorem 3.3(i))
heinlein_ihringer_semidefinite_programming
$$A_2( 7, 4) \le 388 \\ A_2( 8, 3) \le 9191 \\ A_2( 8, 4) \le 6479 \\ A_2( 8, 5) \le 327 \\ A_2( 8, 6) \le 260 \\ A_2( 9, 3) \le 107419 \\ A_2( 9, 4) \le 53710 \\ A_2( 9, 5) \le 2458 \\ A_2( 9, 6) \le 1240 \\ A_2( 10, 3) \le 2531873 \\ A_2( 10, 4) \le 1705394 \\ A_2( 10, 5) \le 48255 \\ A_2( 10, 6) \le 38455 \\ A_2( 10, 7) \le 1219 \\ A_2( 10, 8) \le 1090 \\ A_2( 11, 3) \le 57201557 \\ A_2( 11, 4) \le 28600778 \\ A_2( 11, 5) \le 660265 \\ A_2( 11, 6) \le 330133 \\ A_2( 11, 7) \le 8844 \\ A_2( 11, 8) \le 4480 \\ A_2( 12, 3) \le 2685948795 \\ A_2( 12, 4) \le 1816165540 \\ A_2( 12, 5) \le 26309023 \\ A_2( 12, 6) \le 21362773 \\ A_2( 12, 7) \le 314104 \\ A_2( 12, 8) \le 279476 \\ A_2( 12, 9) \le 4483 \\ A_2( 12, 10) \le 4226 \\ A_2( 13, 3) \le 119527379616 \\ A_2( 13, 4) \le 59763689822 \\ A_2( 13, 5) \le 688127334 \\ A_2( 13, 6) \le 344063682 \\ A_2( 13, 7) \le 4678401 \\ A_2( 13, 8) \le 2343888 \\ A_2( 13, 9) \le 34058 \\ A_2( 13, 10) \le 17133 \\ A_2( 13, 11) \le 259 \\ A_2( 14, 3) \le 11215665059647 \\ A_2( 14, 4) \le 7496516673358 \\ A_2( 14, 5) \le 54724534275 \\ A_2( 14, 6) \le 43890879895 \\ A_2( 14, 7) \le 330331546 \\ A_2( 14, 8) \le 292988615 \\ A_2( 14, 9) \le 2298622 \\ A_2( 14, 10) \le 2164452 \\ A_2( 14, 11) \le 17155 \\ A_2( 14, 12) \le 16642 \\ A_3( 6, 3) \le 967 \\ A_3( 6, 4) \le 788 \\ A_3( 7, 3) \le 15394 \\ A_3( 7, 4) \le 7696 \\ A_3( 7, 5) \le 166 \\ A_3( 8, 3) \le 760254 \\ A_3( 8, 4) \le 627384 \\ A_3( 8, 5) \le 7222 \\ A_3( 8, 6) \le 6727 \\ A_3( 9, 3) \le 34143770 \\ A_3( 9, 4) \le 17071886 \\ A_3( 9, 5) \le 123535 \\ A_3( 9, 6) \le 61962 \\ A_3( 9, 7) \le 490 \\ A_3( 10, 3) \le 5026344026 \\ A_3( 10, 4) \le 4112061519 \\ A_3( 10, 5) \le 16008007 \\ A_3( 10, 6) \le 14893814 \\ A_3( 10, 7) \le 61002 \\ A_3( 10, 8) \le 59539 \\ A_3( 11, 3) \le 675225312722 \\ A_3( 11, 4) \le 337612656529 \\ A_3( 11, 5) \le 818518696 \\ A_3( 11, 6) \le 409259348 \\ A_3( 11, 7) \le 1076052 \\ A_3( 11, 8) \le 539351 \\ A_3( 11, 9) \le 1462 \\ A_3( 12, 3) \le 298950313257852 \\ A_3( 12, 4) \le 244829520433920 \\ A_3( 12, 5) \le 320387589445 \\ A_3( 12, 6) \le 298571221318 \\ A_3( 12, 7) \le 400831735 \\ A_3( 12, 8) \le 391178436 \\ A_3( 12, 9) \le 537278 \\ A_3( 12, 10) \le 532903 \\ A_4( 6, 3) \le 4772 \\ A_4( 6, 4) \le 4231 \\ A_4( 7, 3) \le 142313 \\ A_4( 7, 4) \le 71156 \\ A_4( 7, 5) \le 516 \\ A_4( 8, 3) \le 20482322 \\ A_4( 8, 4) \le 18245203 \\ A_4( 8, 5) \le 68117 \\ A_4( 8, 6) \le 66054 \\ A_4( 9, 3) \le 2341621613 \\ A_4( 9, 4) \le 1170810807 \\ A_4( 9, 5) \le 2132181 \\ A_4( 9, 6) \le 1067796 \\ A_4( 9, 7) \le 2052 \\ A_4( 10, 3) \le 1343547758223 \\ A_4( 10, 4) \le 1194101275238 \\ A_4( 10, 5) \le 1122729102 \\ A_4( 10, 6) \le 1088550221 \\ A_4( 10, 7) \le 1058831 \\ A_4( 10, 8) \le 1050630 \\ A_4( 11, 3) \le 614496020025690 \\ A_4( 11, 4) \le 307248010015067 \\ A_4( 11, 5) \le 140323867490 \\ A_4( 11, 6) \le 70161933745 \\ A_4( 11, 7) \le 33669242 \\ A_4( 11, 8) \le 16847095 \\ A_4( 11, 9) \le 8196 \\ A_5( 6, 3) \le 17179 \\ A_5( 6, 4) \le 15883 \\ A_5( 7, 3) \le 821170 \\ A_5( 7, 4) \le 410585 \\ A_5( 7, 5) \le 1254 \\ A_5( 8, 3) \le 277100135 \\ A_5( 8, 4) \le 256754528 \\ A_5( 8, 5) \le 398154 \\ A_5( 8, 6) \le 391883 \\ A_5( 9, 3) \le 64262978412 \\ A_5( 9, 4) \le 32131489207 \\ A_5( 9, 5) \le 19675409 \\ A_5( 9, 6) \le 9847885 \\ A_5( 9, 7) \le 6254 \\ A_5( 10, 3) \le 108238287449582 \\ A_5( 10, 4) \le 100215014898311 \\ A_5( 10, 5) \le 31196584033 \\ A_5( 10, 6) \le 30703887393 \\ A_5( 10, 7) \le 9803150 \\ A_5( 10, 8) \le 9771883 \\ A_7( 6, 3) \le 123239 \\ A_7( 6, 4) \le 118347 \\ A_7( 7, 3) \le 11807778 \\ A_7( 7, 4) \le 5903889 \\ A_7( 7, 5) \le 4806 \\ A_7( 8, 3) \le 14753449680 \\ A_7( 8, 4) \le 14176726504 \\ A_7( 8, 5) \le 5803270 \\ A_7( 8, 6) \le 5769615 \\ A_7( 9, 3) \le 9728400942608 \\ A_7( 9, 4) \le 4864200471305 \\ A_7( 9, 5) \le 566262547 \\ A_7( 9, 6) \le 283240686 \\ A_7( 9, 7) \le 33618 \\ A_7( 10, 3) \le 85039309360944189 \\ A_7( 10, 4) \le 81703574152063079 \\ A_7( 10, 5) \le 4784663914039 \\ A_7( 10, 6) \le 4756893963688 \\ A_7( 10, 7) \le 282744208 \\ A_7( 10, 8) \le 282508875$$
HeinleinIhringer2018
johnson_MDC_Lemma_2
$$A_2(10,5) \le 48104$$
HonoldKiermaierKurz2018 (Lemma 2)
johnson_MDC_Lemma_3
$$A_3(9,5) \le 123048$$
HonoldKiermaierKurz2018 (Lemma 3)
johnson_MDC_Lemma_5
$$A_q(7,3) \le 2(q^8+q^6+2q^5+2q^3+q^2-q+2)$$
HonoldKiermaierKurz2018 (Lemma 5)
johnson_MDC_Proposition_5
$$A_2(8,3) \le 9260$$ and for $$3 \le q$$: $$A_q(8,3) \le q^{12}+3q^{10}+q^{9}+3q^{8}+3q^{7}+3q^{6}+5q^{5}+3q^{4}+q^{3}+4q^{2}+2q-1$$
HonoldKiermaierKurz2018 (Proposition 5)
nodd_deqn
If $$d=n$$ then the whole vector space is the direct sum of each pair of codewords. If a code had three codewords, then $$2k=n$$ which is impossible for $$n$$ odd.

nodd_deqnm2_u
$$A_q(n,n-2) \le 2q^{k+1}+2$$ for $$n=2k+1 \ge 5$$
HonoldKiermaierKurz20163 (Upper bound of Theorem 3.3.ii)
semidefinite_programming
$$A_2(4,3) \le 6 \\A_2(5,3) \le 20 \\A_2(6,3) \le 124 \\A_2(7,3) \le 776 \\A_2(7,5) \le 35 \\A_2(8,3) \le 9268 \\A_2(8,5) \le 360 \\A_2(9,3) \le 107419 \\A_2(9,5) \le 2485 \\A_2(10,3) \le 2532929 \\A_2(10,5) \le 49394 \\A_2(10,7) \le 1223 \\A_2(11,5) \le 660285 \\A_2(11,7) \le 8990 \\A_2(12,7) \le 323374 \\A_2(12,9) \le 4487 \\A_2(13,7) \le 4691980 \\A_2(13,9) \le 34306 \\A_2(14,9) \le 2334086 \\A_2(14,11) \le 17159 \\A_2(15,11) \le 134095 \\A_2(16,13) \le 67079$$
BachocPassuelloVallentin2013
special_cases_upper_notderived
$$A_2(6,3) \le 118$$ and $$A_2(7,4) \le 407$$
HonoldKiermaierKurz20163 (Theorem 4.1)
special_cases_upper_notderived_2
$$A_2(6,3) \le 117$$, $$A_2(8,4) \le 6479$$, and $$A_2(8,5) \le 326$$
HeinleinKurz2018 (Proposition 3, Proposition 4, and Proposition 5)
trivial_3
A subspace code is a subset of the subspaces of $$\mathbb{F}_q^n$$. So $$A_q(n,d) \le \sum_{k=0}^n \binom{n}{k}_q$$

### upper bound, recursive

Ahlswede_Aydinian_ilp
$$A_q(n,2t+1) \leq \max(\sum_{i=1}^{n}f_{i})\text{ subject to}\\ f_{0}, f_{1}, \dots, f_{n} \text{ nonnegative integers} \\ f_{0}=f_{n}=1, \, f_{k}=f_{n-k}=0 \text{ for } k=1, \dots, t \\ f_{k}\leq A_{q}(n, 2t+2, k) \text{ for } k=0, \dots, n\\ f_{k}+{1\over t+1}\sum_{i=1}^{t}(t+1-i)(f_{k-i}\binom{n-k+i}{n-k}_{q}+f_{k+i}\binom{k +i}{k}_{q}) \leq \binom{n}{k}_{q} \text{ for } k=0, \dots, n\\ f_{-j}=f_{n+j}=0 \text{ for } i=1, \dots, t \text{ (by convention),}\\ \text{for integers }1\leq t \leq {n\over 2}$$
AhlswedeAydinian2009 (Theorem 5 and Theorem 6), KhaleghiSilvaKschischang2009 (Theorem10)
Etzion_Vardy_ilp
$$A_q(n,2e+1) \le \max \sum_{k=0}^n D_k$$ subject to $$D_k \le A_q(n,2e+2,k)$$ for $$k = 0,1, \ldots,n$$ and $$\sum_{i=-e}^e c(k,k+i,e) D_{k+i} \le \binom{n}{k}_{q}$$ for $$k = 0,1, \ldots,n$$ with $$c(j,k,r) = \sum^{\min\{j,k\}}_{i=\lceil {{k+j-r\over 2}}\rceil} \binom{k}{i}_{q} \binom{n-k}{j-i}_{q}q^{(j-i)(k-i)}$$.
EtzionVardy2011 (Theorem 10)
HKK_theorem_3_3_i_upper_bound_mdc
If $$n=2k \ge 8$$ even then $$A_q(n,n-2)=A_q(n,n-2,k)$$.
HonoldKiermaierKurz20163 (Theorem 3.3.i), HeinleinHonoldKiermaierKurzWassermann2017 (Theorem 2)
cdc_upper_bound
$$A_q(n,d) \le \sum_{k=0}^n A_q(n,d;k)$$

improved_cdc_upper_bound
$$\sum_{k=0 \land k \equiv \lfloor n / 2 \rfloor \pmod{d}}^{n} A_q(n,2\lceil d/2 \rceil;k) \le A_q(n,d) \le 2 + \sum_{k=\lceil d/2 \rceil}^{n- \lceil d/2 \rceil} A_q(n,2\lceil d/2 \rceil;k)$$
HonoldKiermaierKurz20163 (Upper bound of Theorem 2.5)
johnson_MDC_Lemma_4
For odd $$v\ge 7$$ we have \begin{eqnarray*} A_q(v,v-4)&\le& \max\Big\{ 2A_q(v,v-3;m-1)+2A_q(v,v-3;m),\\ && 2+2\left\lfloor\left(\binom{2m+1}{1}_{q}-\binom{m-2}{1}_{q}\right)/\binom{m-1}{1}_{q}\right\rfloor\\ &&+2\left\lfloor\left(\binom{2m+1}{1}_{q}-\binom{m-2}{1}_{q}\right)\cdot A_q(2m,2m-2;m-1)/\binom{m}{1}_{q}\right\rfloor \Big\}, \end{eqnarray*} where $$m=(v-1)/2$$.
HonoldKiermaierKurz2018 (Lemma 4)
johnson_MDC_Lemma_6
Let $$m\ge 4$$. If $$A_q(2m,2m-4)>2+A_q(2m,2m-4;m)$$, then we have \begin{align*} A_q(2m,2m-4)\le \left\lfloor \frac{\binom{2m}{1}_{q}}{\binom{m}{1}_{q}}\cdot \left\lfloor \frac{\left(\binom{2m-1}{1}_{q}-\binom{m-3}{1}_{q}\right) \cdot A_q(2m-2,2m-4;m-2)}{\binom{m-1}{1}_{q}} \right\rfloor +\frac{2\binom{2m}{1}_{q}}{\binom{m-2}{1}_{q}} \right\rfloor \end{align*} if $$m=4$$ or $$m=5$$ and $$q=2$$ and \begin{align*} A_q(2m,2m-4)\le \left\lfloor\frac{\binom{2m}{1}_{q}}{\binom{m}{1}_{q}}\cdot \left\lfloor \frac{\binom{2m-1}{1}_{q}\cdot A_q(2m-2,2m-4;m-2)}{\binom{m-1}{1}_{q}} \right\rfloor\right\rfloor \end{align*} otherwise.
HonoldKiermaierKurz2018 (Lemma 6)
relax_d
$$A_q(n,d) \le A_q(n,d-1)$$

special_improved_cdc_upper_bound
$$A_q(n,d) \le \sum_{k=\lceil d/2 \rceil}^{n- \lceil d/2 \rceil} A_q(n,2\lceil d/2 \rceil;k)$$ if $$\lceil d/2 \rceil$$ divides $$v$$ and $$2\le d$$
HonoldKiermaierKurz2018 (Lemma 1)

### lower bound, recursive

HKK_theorem_3_3_i_lower_bound_mdc
If $$n=2k \ge 8$$ even then $$A_q(n,n-2)=A_q(n,n-2,k)$$.
HonoldKiermaierKurz20163 (Theorem 3.3.i), HeinleinHonoldKiermaierKurzWassermann2017 (Theorem 2)
cdc_average_argument
$$\max_{k=0}^n \frac{q^{n+1-k}+q^k-2}{q^{n+1}-1} \cdot A_q(n+1,d+1;k) \le A_q(n,d)$$
EtzionSilberstein2009
cdc_lower_bound
$$\max_{k=0}^n A_q(n,d;k) \le A_q(n,d)$$

improved_cdc_lower_bound
$$\sum_{k=0 \land k \equiv \lfloor n / 2 \rfloor \pmod{d}}^{n} A_q(n,2\lceil d/2 \rceil;k) \le A_q(n,d) \le 2 + \sum_{k=\lceil d/2 \rceil}^{n- \lceil d/2 \rceil} A_q(n,2\lceil d/2 \rceil;k)$$
HonoldKiermaierKurz20163 (Lower bound of Theorem 2.5)
layer_construction
The bound is $$A_q(n,d) \ge \max\{ \sum_{k \in K} A_q(n,d;k) \mid K \subseteq \{0,\ldots,n\} : |k_1-k_2| \ge d \;\forall k_1 \ne k_2 \in K \}$$. This is computed using dynamic programming and the function $$L(N) := \max\{\sum_{k \in K} A_q(n,d;k) \mid K \subseteq \{0,\ldots,N\} : |k_1-k_2| \ge d \;\forall k_1 \ne k_2 \in K\} = \max\{ L(N-1), L(N-d) + A_q(n,d;N) \}$$ for all $$N = 0,\ldots,n$$.
HonoldKiermaierKurz20163