Nonexistence of nontrivial tight 8-designs

Abstract

Tight t-designs are t-designs whose sizes achieve the Fisher type lower bound. We give a new necessary condition for the existence of nontrivial tight designs and then use it to show that there do not exist nontrivial tight 8-designs.

Appendices

A The thirteen degree polynomial

$f_4(v, k) := -16384 k^{12} v+65536 k^{12}+98304 k^{11} v^2-393216 k^{11} v-253952 k^{10} v^3+786432 k^{10} v^2+1744896 k^{10} v-3309568 k^{10}+368640 k^9 v^4-327680 k^9 v^3-8724480 k^9 v^2+16547840 k^9 v-328320 k^8 v^5-1102464 k^8 v^4+17194752 k^8 v^3-21567744 k^8 v^2-49810560 k^8 v+62323584 k^8+182784 k^7 v^6+2050560 k^7 v^5-16432128 k^7 v^4-13016064 k^7 v^3+199242240 k^7 v^2-249294336 k^7 v-61184 k^6 v^7-1642240 k^6 v^6+6536960 k^6 v^5+58253568 k^6 v^4-293538048 k^6 v^3+209662720 k^6 v^2+511604992 k^6 v-488998144 k^6+10752 k^5 v^8+698880 k^5 v^7+1258752 k^5 v^6-59703552 k^5 v^5+183266304 k^5 v^4+243542016 k^5 v^3-1534814976 k^5 v^2+1466994432 k^5 v-640 k^4 v^9-143664 k^4 v^8-2296192 k^4 v^7+27050224 k^4 v^6-7038496 k^4 v^5-582955856 k^4 v^4+1856597696 k^4 v^3-1428764528 k^4 v^2-1015706784 k^4 v+974873344 k^4+7520 k^3 v^9+772608 k^3 v^8-2875616 k^3 v^7-58917568 k^3 v^6+469164960 k^3 v^5-1155170432 k^3 v^4+412538336 k^3 v^3+2031413568 k^3 v^2-1949746688 k^3 v+336 k^2 v^{10}-52816 k^2 v^9-1582560 k^2 v^8+27560816 k^2 v^7-127930016 k^2 v^6+28759472 k^2 v^5+1497511456 k^2 v^4-4944873072 k^2 v^3+6922441360 k^2 v^2-4733985888 k^2 v+1506333312 k^2-2352 k v^{10}+203472 k v^9-764688 k v^8-24513072 k v^7+293023248 k v^6-1459281552 k v^5+3929166288 k v^4-5947568016 k v^3+4733985888 k v^2-1506333312 k v+45 v^{11}+972 v^{10}-191952 v^9+2961396 v^8-14780538 v^7-18769932 v^6+544096980 v^5-2755473732 v^4+7281931941 v^3-11097146016 v^2+9310949028 v-3408102864$.

B Proof of Proposition 5.3

Lemmas B.1, B.2, B.3, and B.4, prove Proposition 5.3. The proofs of these lemmas are not difficult once the appropriate auxiliary polynomials g has been explicitly identified. Such auxiliary polynomials were obtained through an ad-hoc procedure that we will not describe here.

Lemma B.1

If $v \in [2k, 9k]$ and $k \ge 20000$, then $f_4(v, k) > 0$.

Proof

Let

$$\begin{aligned} g(v, k):= & {} 65535 k^{12} - 16404 k^{12} v + 98304 k^{11} v^2 - 253969 k^{10} v^3 \\&+\, 368584 k^9 v^4 - 328320 k^8 v^5 + 182701 k^7 v^6 - 61184 k^6 v^7 \\&+\,10744 k^5 v^8 - 640 k^4 v^9 + 335 k^2 v^{10} + 45 v^{11}. \end{aligned}$$

View $f_4(v, k)$ and g(v, k) as polynomials in v with coefficients in ${{\mathrm{\mathbb {R}}}}[k]$. Using the fact that $k \ge 20000$, we can check that for every $i \ge 0$, the coefficient of $v^i$ in $f_4(v, k)$ is no smaller than that in g(v, k), and for some i, the coefficient of $v^i$ in $f_4(v, k)$ is strictly larger than that in g(v, k). So, we find that $f_4(v, k) > g(v, k)$ since $v \ge 2 k > 0$.

Let $t := v / k$, so that $t \in [2, 9]$. We have

$$\begin{aligned} f_4(v, k) > g(v, k)= & {} g(t k, k) \\= & {} h_{13}(t) k^{13} + h_{12}(t) k^{12} + h_{11}(t) k^{11}, \end{aligned}$$

where

$$\begin{aligned} h_{13}(t)&:= -16404 t + 98304 t^2 - 253969 t^3 + 368584 t^4 \\&\quad - 328320 t^5 + 182701 t^6 - 61184 t^7 + 10744 t^8 - 640 t^9, \\ h_{12}(t)&:= 65535 + 335 t^{10}, \\ h_{11}(t)&:= 45 t^{11}. \end{aligned}$$

One verifies that $h_{13}$, $h_{12}$ and $h_{11}$ are positive when $t \in [2, 9]$, and then the result follows. $\square$

Recall the definitions of real numbers a and b in Eq. (5.1):

$$\begin{aligned} {\left\{ \begin{array}{ll} a = \frac{2}{1 - \root 4 \of {\frac{3}{8}}} = \frac{2}{5} (8 + 2 \sqrt{6} + \sqrt{48 + 22 \sqrt{6}}) \approx 9.1971905725, \\ b = \frac{23}{500} \left( 249 + 86 \sqrt{6} + \sqrt{171312 + 70918 \sqrt{6}} \right) \approx 48.1640392521. \\ \end{array}\right. } \end{aligned}$$

Lemma B.2

If $v \in [9 k, a k + b]$ and $k \ge 100$, then $f_4(v, k) > 0$.

Proof

Let $t := a k + b - v$, so that $t \in [0, (a - 9) k + b] \subseteq [0, 0.9 k]$ when $k \ge 100$. Let

$$\begin{aligned} g(t, k)&:= 13700000000 t k^{12} - 13400000000 t^2 k^{11} - 140000000 t^3 k^{10} \\&\quad - 1380000000 t^4 k^9 - 6000000 t^5 k^8 - 21000000 t^6 k^7 \\&\quad - 10000 t^7 k^6 - 73000 t^8 k^5 - 40000 t^9 k^3 - 7 t^9 k^4 \\&\quad - 50000 t^{10} k - 200 t^{10} k^2 - 312000000000 t^{11}. \end{aligned}$$

View $f_4(a k + b - t, k)$ and g(t, k) as polynomials in k with coefficients in ${{\mathrm{\mathbb {R}}}}[t]$. Using the fact that $t \ge 0$, we can check that for every $i \ge 0$, the coefficient of $k^i$ in $f_4(a k + b - t, k)$ is no smaller than that in g(t, k), and for some i, the coefficient of $k^i$ in $f_4(a k + b - t, k)$ is strictly larger than that in g(t, k). So, we find that $f_4(a k + b - t, k) > g(t, k)$ since $k > 0$.

Let $s := t / k$, so that $s \in [0, 0.9]$. We have

$$\begin{aligned} f_4(v, k)= & {} f_4(a k + b - t) > g(t, k) = g(s k, k) \\= & {} h_{13}(s) k^{13} - h_{12}(s) k^{12} - h_{11}(s) k^{11}, \end{aligned}$$

where

$$\begin{aligned} h_{13}(s)&:= 13700000000 s - 13400000000 s^2 - 140000000 s^3 \\&\quad - 1380000000 s^4 - 6000000 s^5 - 21000000 s^6 - 10000 s^7 \\&\quad - 73000 s^8 - 7 s^9, \\ h_{12}(s)&:= 40000 s^9 + 200 s^{10}, \\ h_{11}(s)&:= 50000 s^{10} + 312000000000 s^{11}. \end{aligned}$$

One verifies that $h_{13}$, $h_{12}$ and $h_{11}$ are positive when $s \in [0, 0.9]$. Since $k \ge 100$

$$\begin{aligned} f_4(v, k)&> h_{13}(s) k^{13} - h_{12}(s) k^{12} - h_{11}(s) k^{11} \\&\ge (h_{13}(s) - h_{12}(s) / 100 - h_{11}(s) / 100^2) k^{13} \\&= \bigg (13700000000 s - 13400000000 s^2 - 140000000 s^3 \\&\quad - 1380000000 s^4 - 6000000 s^5 - 21000000 s^6 - 10000 s^7 \\&\quad - 73000 s^8 - 407 s^9 - 7 s^{10} - 31200000 s^{11}\bigg ) k^{13}. \end{aligned}$$

The result follows from the fact that the right-hand side of the expression above is positive when $s \in [0, 0.9]$.

Lemma B.3

If $a k + b + \frac{1}{100} \le v \le 10 k$ and $k \ge 10^5$, then $f_4(v, k) < 0$.

Proof

Let $t := v - a k - b$, so that $t \in [\frac{1}{100}, (10 - a) k - b] \subseteq [\frac{1}{100}, k]$ when $k \ge 10^5$. Let

$$\begin{aligned} g(t, k)&:= -137892000 k^{12} + 13318642886180 k^{11} + 713748202323829 k^{10} \\&\quad + 48837673261525668 k^9 + 5098485316801241991 k^8 \\&\quad + 980132640412645508268 k^7 + 488910709935302976594934 k^6 \\&\quad + 1305009906289977795675277621 k^5 \\&\quad + 151418572251274917743210971453210 k^4 \\&\quad + 64000 t^9 k^3 + 2000 t^{10} k^2 + 7000 t^{10} k + 127 t^{11}. \end{aligned}$$

View $f_4(a k + b + t, k)$ and g(t, k) as polynomials in k with coefficients in ${{\mathrm{\mathbb {R}}}}[t]$. Using the fact that $t \ge \frac{1}{100}$, we can check that for every $i \ge 0$, the coefficient of $k^i$ in $f_4(a k + b + t, k)$ is no larger than that in g(t, k), and for some i, the coefficient of $k^i$ in $f_4(a k + b + t, k)$ is strictly smaller than that in g(t, k). So, we find that $f_4(a k + b + t, k) < g(t, k)$ since $k > 0$. It follows from $t \le k$ that $g(t, k) \le g(k, k)$. It is easy to show the one variable polynomial g(k, k) takes negative value when $k \ge 10^5$. Thus, $f_4(v, k)< g(t, k) \le g(k, k) < 0$. $\square$

Lemma B.4

If $v \in [9.24 k, 0.8 k^2]$ and $k \ge 10^5$, then $f_4(v, k) < 0$.

Proof

Let

$$\begin{aligned} g(v, k)&:= 65536 k^{12} - 16384 k^{12} v + 98312 k^{11} v^2 - 253952 k^{10} v^3 \\&\quad + 368640 k^9 v^4 - 328299 k^8 v^5 + 182784 k^7 v^6 - 61177 k^6 v^7 \\&\quad + 10752 k^5 v^8 - 639 k^4 v^9 + 336 k^2 v^{10} + 45 v^{11}. \end{aligned}$$

View $f_4(v, k)$ and g(v, k) as polynomials in v with coefficients in ${{\mathrm{\mathbb {R}}}}[k]$. Using the fact that $k \ge 10^5$, we can check that for every $i \ge 0$, the coefficient of $v^i$ in $f_4(v, k)$ is no larger than that in g(v, k), and for some i, the coefficient of $v^i$ in $f_4(v, k)$ is strictly smaller than that in g(v, k). So, we find that $f_4(v, k) < g(v, k)$ since $v > 0$.

Let $t := v / k$, so that $t \in [9.24, 0.8k]$. Let

$$\begin{aligned} h(x)&:= - 1 + 16384 x - 98312 x^2 + 253952 x^3 - 368640 x^4 \\&\quad + 328299 x^5 - 182784 x^6 + 61177 x^7 - 10752 x^8 + 639 x^9. \end{aligned}$$

We have

$$\begin{aligned}&f_4(v, k) < g(v, k) = g(t k, k) \nonumber \\ =&- \big (k - 65536\big ) k^{12} - \big (h(t) - 298 t^9\big ) k^{13} \nonumber \\&- \big (298 - 336 (t / k) - 45 (t / k)^2\big ) t^9 k^{13} \end{aligned}$$

(B.1)

$$\begin{aligned} =&- \big (k - 65536\big ) k^{12} - \big (h(t) - 0.1 t^9\big ) k^{13} \nonumber \\&- \big (0.1 - 336 (t / k) - 45 (t / k)^2\big ) t^9 k^{13}. \end{aligned}$$

(B.2)

It suffices to prove that the coefficients of $k^{12}$, $k^{13}$ and $t^9 k^{13}$ in Eq. (B.1) are all negative, or those coefficients in Eq. (B.2) are all negative. Since $k \ge 10^5$, we have $0.00026 k^2 \ge 26 k$. So, either $v \ge 26 k$ or $v \le 0.00026 k^2$.

Case 1: $v \in [26 k, 0.8 k^2]$, so that $t \in [26, 0.8 k]$ and $t / k \in [0, 0.8]$.

Consider Eq. (B.1). The result follows from the following facts:

• (i) $k - 65536 \ge 0$;

• (ii) $h(t) - 298 t^9 \ge 0$ when $t \ge 26$;

• (iii) $298 - 336 (t / k) - 45 (t / k)^2 \ge 0$ when $t / k \in [0, 0.8]$.

Case 2: $v \in [9.24k, 0.00026 k^2]$ so that $t \in [9.24, 0.00026 k]$ and $t / k \in [0, 0.00026]$.

Consider Eq. (B.2). The result follows from the following facts:

• (i) $k - 65536 \ge 0$;

• (ii) $h(t) - 0.1 t^9 \ge 0$ when $t \ge 9.24$;

• (iii) $0.1 - 336 (t / k) - 45 (t / k)^2 \ge 0$ when $t / k \in [0, 0.00026]$. $\square$