New results on quaternary codes and their Gray map images for constructing uniform designs

Abstract

The research of developing efficient methodologies for constructing optimal experimental designs has been very active in the last decade. Uniform design is one of the most popular approaches, carried out by filling up experimental points in a determinately uniform fashion. Applications of coding theory in experimental design are interesting and promising. Quaternary codes and their binary Gray map images attracted much attention from those researching design of experiments in recent years. The present paper aims at exploring new results for constructing uniform designs based on quaternary codes and their binary Gray map images. This paper studies the optimality of quaternary designs and their two and three binary Gray map image designs in terms of the uniformity criteria measured by: the Lee, wrap-around, symmetric, centered and mixture discrepancies. Strong relationships between quaternary designs and their two and three binary Gray map image designs are obtained, which can be used for efficiently constructing two-level designs from four-level designs and vice versa. The significance of this work is evaluated by comparing our results to the existing literature.

Appendix

Proof of Theorem 1

The proof is obvious by the same technique as that in Elsawah and Qin (2015a, b). $\square$

Proof of Theorem 2

For any quaternary design $\mathcal {U}_q\in \mathbb {U}(n; 4^m),$ let $\beta _{ij},~\alpha _{ij},~\kappa _{ij},~\rho _{ij}, \ell _{ij}$ and $\gamma _{ij}$ as given in Theorem 1. Then, we have

(a):

$\beta _{ij}+\alpha _{ij}+\ell _{ij}+\gamma _{ij}+\kappa _{ij}+\rho _{ij}=m,$

(b):

$\sum _{i=1}^{n}\sum _{j\ne i}^{n}\beta _{ij}=\sum _{i=1}^{n}\sum _{j\ne i}^{n}\alpha _{ij}=\frac{1}{8}mn(n-4),$

(c):

$\sum _{i=1}^{n}\beta _{ii}=\sum _{i=1}^{n}\alpha _{ii}=\frac{1}{2}mn,~\beta _{ii}+\alpha _{ii}=m,~\ell _{ii}=\gamma _{ii}=\kappa _{ii}=\rho _{ii}=0,$

(d):

$\sum _{i=1}^{n}\sum _{j\ne i}^{n}\ell _{ij}=\sum _{i=1}^{n}\sum _{j\ne i}^{n}\gamma _{ij}=2\sum _{i=1}^{n}\sum _{j\ne i}^{n}\kappa _{ij}=2\sum _{i=1}^{n}\sum _{j\ne i}^{n}\rho _{ij}=\frac{1}{4}mn^2.$

Thus, we can rewrite the analytical expressions of the uniformity criteria in Theorem 1 as follows

Table 12 Distributions of $\zeta _{l},~\zeta _{w},~\zeta _{c},~\zeta _{m},~\zeta ^*_{c}$ and $\zeta ^*_{m}$-values for $\mathcal {U}_q\in \mathbb {U}(n; 4^m)$

$$\begin{aligned}{}[{\mathcal {LD}}(\mathcal {U}_q)]^2 =&-\left( \frac{3}{4}\right) ^{m}+\frac{1}{n}+\frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1,i\ne j}^{n}\left( \frac{3}{4}\right) ^{\rho _{ij}+\kappa _{ij}+\ell _{ij}}\left( \frac{1}{2}\right) ^{\gamma _{ij}}. \end{aligned}$$

(7.1)

$$\begin{aligned}{}[{\mathcal {WD}}(\mathcal {U}_q)]^2=&-\left( \frac{4}{3}\right) ^{m}+\frac{1}{n}\left( \frac{3}{2}\right) ^{m}+\frac{1}{n^{2}}\left( \frac{5}{4}\right) ^{m}\sum _{i=1}^{n}\sum _{j=1,i\ne j}^{n}\left( \frac{6}{5}\right) ^{\beta _{ij}+\alpha _{ij}}\left( \frac{21}{20}\right) ^{\rho _{ij}+\kappa _{ij}+\ell _{ij}}. \end{aligned}$$

(7.2)

$$\begin{aligned}{}[{\mathcal {CD}}(\mathcal {U}_q)]^2=&\left( \frac{13}{12}\right) ^{m}-\frac{2}{n}\sum _{i=1}^{n}\left( \frac{143}{128}\right) ^{\beta _{ii}}\left( \frac{135}{128}\right) ^{\alpha _{ii}}+\frac{1}{n^{2}}\left( \frac{9}{8}\right) ^{m}\sum _{i=1}^{n}\left( \frac{11}{9}\right) ^{\beta _{ii}}\nonumber \\&+\frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1,i\ne j}^{n}\left( \frac{11}{8}\right) ^{\beta _{ij}}\left( \frac{9}{8}\right) ^{\alpha _{ij}+\ell _{ij}}. \end{aligned}$$

(7.3)

$$\begin{aligned}{}[{\mathcal {MD}}(\mathcal {U}_q)]^2 =&\left( \frac{19}{12}\right) ^{m}-\frac{2}{n}\sum _{i=1}^{n}\left( \frac{1181}{768}\right) ^{\beta _{ii}}\left( \frac{1253}{768}\right) ^{\alpha _{ii}}+\frac{1}{n^{2}}\left( \frac{27}{16}\right) ^{m}\sum _{i=1}^{n}\left( \frac{29}{27}\right) ^{\alpha _{ii}}\nonumber \\&+\frac{1}{n^{2}}\sum _{i=1}^{n}\sum _{j=1,i\ne j}^{n}\left( \frac{53}{32}\right) ^{\kappa _{ij}}\left( \frac{45}{32}\right) ^{\rho _{ij}}\left( \frac{3}{2}\right) ^{\gamma _{ij}}\left( \frac{51}{32}\right) ^{\ell _{ij}}\left( \frac{27}{16}\right) ^{\beta _{ij}}\left( \frac{29}{16}\right) ^{\alpha _{ij}}. \end{aligned}$$

(7.4)

Define $\zeta _{l}=1-\min \left\{ \frac{|x_{ik}-x_{jk}|}{4},1-\frac{|x_{ik}-x_{jk}|}{4}\right\} ,$ $\zeta _{w}=\frac{3}{2}-|y_{ik}-y_{jk}|(1-|y_{ik}-y_{jk}|), ~\zeta _{c}=1+\frac{1}{2}|y_{ik}-\frac{1}{2}|+\frac{1}{2}|y_{jk}-\frac{1}{2}|-\frac{1}{2}|y_{ik}-y_{jk}|,~\zeta _{m}=\frac{15}{8}-\frac{1}{4}|y_{ik}-\frac{1}{2}|-\frac{1}{4}|y_{jk}-\frac{1}{2}|-\frac{3}{4}|y_{ik}-y_{jk}|+\frac{1}{2}|y_{ik}-y_{jk}|^2,~ \zeta ^*_{c}=1+\frac{1}{2}|y_{ik}-\frac{1}{2}|-\frac{1}{2}|y_{ik}-\frac{1}{2}|^{2}$ and $\zeta ^*_{m}=\frac{5}{3}-\frac{1}{4}|y_{ik}-\frac{1}{2}|-\frac{1}{4}|y_{ik}-\frac{1}{2}|^{2}.$ Table 12 gives the distributions of $\zeta _{l},~\zeta _{w},~\zeta _{c},~\zeta _{m},~\zeta ^*_{c}$ and $\zeta ^*_{m},$ which are very useful to find the lower bounds of these discrepancies for quaternary designs. Thus, the expressions of the lower bounds of the uniformity criteria are straightforward according to the analytical expressions (7.1)–(7.4), the relations (a)–(d), Table 12 and based on the following two inequalities. Suppose $\sum _{i=1}^{n}x_{i} = c,~\theta =n(\lfloor \frac{c}{n}\rfloor +1)-c,~x_{i}$’s are nonnegative integers, then for any positive $\mu ,$ we have $\sum _{i=1}^{n}\mu ^{x_{i}}\ge \mu ^{\lfloor \frac{c}{n}\rfloor }(\theta +(n-\theta )\mu )$ (cf. Lemma 4 in Elsawah and Qin 2015c). Suppose $\sum _{i=1}^{n}z_{i} = c_1,~\sum _{i=1}^{n}y_{i} = c_2,$ for $i=1,2,\ldots ,n,$ where $z_{i} ,~y_{i}$’s are nonnegative integers and $\alpha _s=n(\lfloor \frac{c_s}{n}\rfloor +1)-c_s,~s=1,2,$ then for any positive $\mu _1$ and $\mu _2,$ we have $\sum _{i=1}^{n}\mu _1^{z_i}\mu _2^{y_i}\ge \left\{ \begin{array}{ll} (n-\alpha _1)e^{\chi _1}+(n-\alpha _2)e^{\chi _2}+(\alpha _1+\alpha _2-n)e^{\chi _3},~ \alpha _1+\alpha _1>n,\\ \alpha _2e^{\chi _1}+\alpha _1e^{\chi _2}+(n-\alpha _1-\alpha _2)e^{\chi _4},~ \alpha _1+\alpha _2\le n, \end{array} \right.$ where $\chi _1=\varepsilon _1(\lfloor \frac{c_1}{n}\rfloor +1)+\varepsilon _2\lfloor \frac{c_2}{n}\rfloor ,~\chi _2=\varepsilon _1\lfloor \frac{c_1}{n}\rfloor +\varepsilon _2(\lfloor \frac{c_2}{n}\rfloor +1),~\chi _3=\varepsilon _1\lfloor \frac{c_1}{n}\rfloor +\varepsilon _2\lfloor \frac{c_2}{n}\rfloor ,~\chi _4=\varepsilon _1(\lfloor \frac{c_1}{n}\rfloor +1)+\varepsilon _2(\lfloor \frac{c_2}{n}\rfloor +1)$ and $(n-\alpha _1)\chi _1+(n-\alpha _2)\chi _2+(\alpha _1+\alpha _2-n)\chi _3=\alpha _2\chi _1+\alpha _1\chi _2+(n-\alpha _2-\alpha _1)\chi _4=\varepsilon _1c_1+\varepsilon _2c_2,~\varepsilon _1=\ln \mu _1$ and $\varepsilon _2=\ln \mu _2$ (cf. Lemma 4 in Elsawah and Qin 2014). Utilizing the same technique as that in Elsawah and Qin (2015b) and Elsawah (2017a) with some algebra, we can complete the proof. $\square$

Table 13 Any pair of points $(x^q_{ik},x^q_{jk})$ in any quaternary design $\mathcal {U}_{q}\in \mathbb {U}(n; 4^{m})$ and their two images $(x^{2bi}_{ik_1},x^{2bi}_{jk_1})$ and $(x^{2bi}_{ik_2},x^{2bi}_{jk_2})$ in the two binary image design $\mathcal {U}_{2bi}\in \mathbb {U}(n; 2^{2m})$

Proof of Theorem 3

The analytical expression $[{\mathcal {VD}^*}(\mathcal {U}_{2bi})]^2$ can be found in Theorem 1 in Elsawah and Qin (2015c). The proof of the analytical expression $[{\mathcal {VD}^{**}}(\mathcal {U}_{2bi})]^2$ is from Lemma 1 in Jiang and Ai (2014) and the discussion in Sect. 3 in Chatterjee et al. (2017) with some algebra, where we extend the existing results from two discrepancies, the $\mathcal {CD}$ and $\mathcal {WD},$ to all the five discrepancies, the $\mathcal {LD},~\mathcal {WD},~\mathcal {SD},~\mathcal {CD}$ and $\mathcal {MD},$ as well as we rewrite the analytical expressions of all these discrepancies as only one analytical expression $[{\mathcal {VD}^{**}}(\mathcal {U}_{2bi})]^2.$ Note that, for this case we have

$$\begin{aligned} aJ_2+a(b-1)I_2=\left( \begin{array}{cc} ab &{} a \\ a &{} ab \\ \end{array} \right) . \end{aligned}$$

However, the proof of the analytical expression $[{\mathcal {VD}^{***}}(\mathcal {U}_{2bi})]^2$ is given as follows. Under the two binary Gray map image principle, any pair of points $(x^q_{ik},x^q_{jk})$ in any quaternary design $\mathcal {U}_{q}\in \mathbb {U}(n; 4^{m})$ will be replaced by the two pairs $(x^{2bi}_{ik_1},x^{2bi}_{jk_1})$ and $(x^{2bi}_{ik_2},x^{2bi}_{jk_2})$ in the two binary Gray map image design $f_2(\mathcal {U}_{q})=\mathcal {U}_{2bi}\in \mathbb {U}(n; 2^{2m})$ as given in Table 13. From Table 13, we can show that

$$\begin{aligned}&\mathcal {N}\left\{ (i,j):(x^{2bi}_{ik},x^{2bi}_{jk})=\left( \frac{3}{4},\frac{3}{4}\right) \right\} =\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{3}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{3}{8}\right) \right\} \\&\quad +\,2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{7}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{7}{8}\right) \right\} .\\&\mathcal {N}\left\{ (i,j):(x^{2bi}_{ik},x^{2bi}_{jk})=\left( \frac{1}{4},\frac{1}{4}\right) \right\} =2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{1}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{3}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{7}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{1}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{3}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{1}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{7}{8}\right) \right\} .\\&\mathcal {N}\left\{ (i,j):(x^{2bi}_{ik},x^{2bi}_{jk})=\left( \frac{3}{4},\frac{1}{4}\right) \right\} =\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{1}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{7}{8}\right) \right\} +2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{1}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{3}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{7}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{1}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{3}{8}\right) \right\} .\\&\mathcal {N}\left\{ (i,j):(x^{2bi}_{ik},x^{2bi}_{jk})=\left( \frac{1}{4},\frac{3}{4}\right) \right\} =\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{3}{8}\right) \right\} \\&\quad +\,2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{7}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{7}{8}\right) \right\} \\&\quad +\,\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{3}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{5}{8}\right) \right\} . \end{aligned}$$

Then, we get

$$\begin{aligned} h_{ij}=\mathcal {N}\left\{ (i,j):(x^{2bi}_{ik},x^{2bi}_{jk})\in \left\{ \left( \frac{1}{4},\frac{3}{4}\right) ,\left( \frac{3}{4},\frac{1}{4}\right) \right\} \right\} =2\gamma _{ij}+\rho _{ij}+\kappa _{ij}+\ell _{ij} \end{aligned}$$

and

$$\begin{aligned} 2m-h_{ij}= & {} \mathcal {N}\left\{ (i,j):(x^{2bi}_{ik},x^{2bi}_{jk})\in \left\{ \left( \frac{1}{4},\frac{1}{4}\right) ,\left( \frac{3}{4},\frac{3}{4}\right) \right\} \right\} \\= & {} 2\beta _{ij}+2\alpha _{ij}+\rho _{ij}+\kappa _{ij}+\ell _{ij}. \end{aligned}$$

Thus, from the analytical expression $[{\mathcal {VD}^*}(\mathcal {U}_{2bi})]^2$ we get

$$\begin{aligned}{}[{\mathcal {VD}}(\mathcal {U}_{2bi})]^2&=\Delta (2m)+\frac{1}{n^{2}}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}(ab)^{2m-h_{ij}}a^{h_{ij}}\right) \nonumber \\&=\Delta (2m)+\frac{1}{n^2}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}(ab)^{2\beta _{ij}+2\alpha _{ij}+\rho _{ij}+\kappa _{ij}+\ell _{ij}}a^{2\gamma _{ij}+\rho _{ij}+\kappa _{ij}+\ell _{ij}}\right) \nonumber \\&=\Delta (2m)+\frac{1}{n^2}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}\left( {a^2}{b^2}\right) ^{\beta _{ij}+\alpha _{ij}}\left( {a^2}{b}\right) ^{\rho _{ij}+\kappa _{ij}+\ell _{ij}}(a^2)^{\gamma _{ij}}\right) . \end{aligned}$$

$\square$

Proof of Theorem 4

For any two binary Gray map image design $\mathcal {U}_{2bi}\in \mathbb {U}(n;2^{2m}),$ let the Hamming distance $h_{ij}=\mathcal {N}\left\{ (i,j):(x_{ik},x_{jk})\in \left\{ \left( \frac{1}{4},\frac{3}{4}\right) ,\left( \frac{3}{4},\frac{1}{4}\right) \right\} \right\} .$ Then, we have

$$\begin{aligned} h_{ii}=0~\text{ and }~\sum _{i=1}^{n}\sum _{j\ne i}^{n}{h_{ij}}=nm(n-2). \end{aligned}$$

(7.5)

The lower bound ${\mathcal {LBV}^*_{2bi}}$ is from the analytical expression $[{\mathcal {VD}^{*}}(\mathcal {U}_{2bi})]^2$ in Theorem 3, the relation (7.5) and Lemma 4 in Elsawah and Qin (2015c). The proof of the lower bound ${\mathcal {LBV}^{**}_{2bi}}$ is from the new analytical expression $[{\mathcal {VD}^{**}}(\mathcal {U}_{2bi})]^2$ in Theorem 3, the proof of Lemma 3 in Chatterjee et al. (2017) and Lemma 2 in Chatterjee et al. (2012) with some algebra, where we extend the existing results from two discrepancies, the $\mathcal {CD}$ and $\mathcal {WD},$ to all the five discrepancies, the $\mathcal {LD},~\mathcal {WD},~\mathcal {SD},~\mathcal {CD}$ and $\mathcal {MD},$ as well as we express the lower bounds of all these discrepancies as only one analytical expression $[{\mathcal {LBV}^{**}}(\mathcal {U}_{2bi})]^2.$ However, the proof of the lower bound ${\mathcal {LBV}^{***}_{2bi}}$ is given as follows. From the relations (a) and (c) in the proof of Theorem 2, we can rewrite the analytical formulas of different discrepancies in $[{\mathcal {VD}^{***}}(\mathcal {U}_{2bi})]^2$ as follows

$$\begin{aligned}{}[{\mathcal {VD}^{***}}(\mathcal {U}_{2bi})]^2= & {} \Delta (2m)+\frac{1}{n^2}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}\left( {a^2}{b^2}\right) ^{\beta _{ij}+\alpha _{ij}}\left( {a^2}{b}\right) ^{\rho _{ij}+\kappa _{ij}+\ell _{ij}}(a^2)^{\gamma _{ij}}\right) \nonumber \\= & {} \Delta (2m)+\frac{1}{n}{{a}^{2m}{b}^{2m}}\nonumber \\&+\,\frac{1}{n^2}\left( \sum _{i=1}^{n}\sum _{j\ne i}^{n}\left( {a^2}{b^2}\right) ^{\beta _{ij}+\alpha _{ij}}\left( {a^2}{b}\right) ^{\rho _{ij}+\kappa _{ij}+\ell _{ij}}(a^2)^{m-\beta _{ij}-\alpha _{ij}-\rho _{ij}-\kappa _{ij}-\ell _{ij}}\right) \nonumber \\= & {} \Delta (2m)+\frac{{a}^{2m}{b}^{2m}}{n}+\frac{1}{n^2}a^{2m}\left( \sum _{i=1}^{n}\sum _{j\ne i}^{n}({b^2})^{\beta _{ij}+\alpha _{ij}}({b})^{\rho _{ij}+\kappa _{ij}+\ell _{ij}}\right) . \end{aligned}$$

(7.6)

From the relations (b) and (d) in the proof of Theorem 2, we get

$$\begin{aligned} \sum _{i=1}^{n}\sum _{j\ne i}^{n}(\rho _{ij}+\kappa _{ij}+\ell _{ij})=\frac{1}{2}mn^2\quad \text{ and }\quad \sum _{i=1}^{n}\sum _{j\ne i}^{n}(\beta _{ij}+\alpha _{ij})=\frac{1}{4}mn(n-4).\nonumber \\ \end{aligned}$$

(7.7)

The proof now is obvious from (7.6) and (7.7) by using Lemma 4 in Elsawah and Qin (2014). $\square$

Proof of Theorem 5

The analytical expression $[{\mathcal {VD}^*}(\mathcal {U}_{3bi})]^2$ can be found in Theorem 1 in Elsawah and Qin (2015c). The proof of the analytical expression $[{\mathcal {VD}^{**}}(\mathcal {U}_{3bi})]^2$ is from Lemma 1 in Jiang and Ai (2014) and the discussion in Sect. 4 in Chatterjee et al. (2017) with some algebra, where we extend the existing results from two discrepancies, the $\mathcal {CD}$ and $\mathcal {WD},$ to all the five discrepancies, the $\mathcal {LD},~\mathcal {WD},~\mathcal {SD},~\mathcal {CD}$ and $\mathcal {MD},$ as well as we rewrite the analytical expressions of all these discrepancies as only one analytical expression $[{\mathcal {VD}^{**}}(\mathcal {U}_{3bi})]^2.$ Note that, for this case we have

$$\begin{aligned} a^3bJ_4+a^3b(b^2-1)I_4=\left( \begin{array}{cccc} a^3b^3 &{} a^3b&{} a^3b&{} a^3b \\ a^3b &{} a^3b^3&{} a^3b&{} a^3b \\ a^3b&{} a^3b&{}a^3b^3&{} a^3b\\ a^3b&{} a^3b&{} a^3b&{}a^3b^3\\ \end{array} \right) . \end{aligned}$$

However, the proof of the analytical expression $[{\mathcal {VD}^{***}}(\mathcal {U}_{3bi})]^2$ is given as follows. Under the three binary Gray map image principle, any pair of points $(x^q_{ik},x^q_{jk})$ in any quaternary design $\mathcal {U}_{q}\in \mathbb {U}(n; 4^{m})$ will be replaced by the three pairs $(x^{3bi}_{ik_1},x^{3bi}_{jk_1})$, $(x^{3bi}_{ik_2},x^{3bi}_{jk_2})$ and $(x^{3bi}_{ik_3},x^{3bi}_{jk_3})$ in the three binary Gray map image design $f_3(\mathcal {U}_{q})=\mathcal {U}_{3bi}\in \mathbb {U}(n; 2^{3m})$ as given in Table 14. From Table 14, we can show that

$$\begin{aligned}&\mathcal {N}\left\{ (i,j):(x^{3bi}_{ik},x^{3bi}_{jk})=\left( \frac{1}{4},\frac{1}{4}\right) \right\} =3\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{1}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{3}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{5}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{7}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{1}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{3}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{1}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{1}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{7}{8}\right) \right\} .\\&\mathcal {N}\left\{ (i,j):(x^{3bi}_{ik},x^{3bi}_{jk})=\left( \frac{3}{4},\frac{3}{4}\right) \right\} =2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{3}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{7}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{3}{8}\right) \right\} +2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{5}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{7}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{3}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{5}{8}\right) \right\} +2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{7}{8}\right) \right\} .\\&\mathcal {N}\left\{ (i,j):(x^{3bi}_{ik},x^{3bi}_{jk})=\left( \frac{1}{4},\frac{3}{4}\right) \right\} =2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{3}{8}\right) \right\} \\&\quad +2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{5}{8}\right) \right\} +2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{1}{8},\frac{7}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{7}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{3}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{3}{7}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{3}{7}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{5}{7}\right) \right\} .\\&\mathcal {N}\left\{ (i,j):(x^{3bi}_{ik},x^{3bi}_{jk})=\left( \frac{3}{4},\frac{1}{4}\right) \right\} =2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{1}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{5}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{3}{8},\frac{7}{8}\right) \right\} \\&\quad +2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{1}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{3}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{5}{8},\frac{7}{8}\right) \right\} +2\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{1}{8}\right) \right\} \\&\quad +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{3}{8}\right) \right\} +\mathcal {N}\left\{ (i,j):(x^{q}_{ik},x^{q}_{jk})=\left( \frac{7}{8},\frac{5}{8}\right) \right\} . \end{aligned}$$

Table 14 Any pair of points $(x^q_{ik},x^q_{jk})$ in any quaternary design $\mathcal {U}_{q}\in \mathbb {U}(n; 4^{m})$ and their three images $(x^{3bi}_{ik_1},x^{3bi}_{jk_1})$, $(x^{3bi}_{ik_2},x^{3bi}_{jk_2})$ and $(x^{3bi}_{ik_3},x^{3bi}_{jk_3})$ in the three binary image design $\mathcal {U}_{3bi}\in \mathbb {U}(n; 2^{3m}).$

Then, we get

$$\begin{aligned} h_{ij}=\mathcal {N}\left\{ (i,j):(x^{3bi}_{ik},x^{3bi}_{jk})\in \left\{ \left( \frac{1}{4},\frac{3}{4}\right) ,\left( \frac{3}{4},\frac{1}{4}\right) \right\} \right\} =2\gamma _{ij}+2\rho _{ij}+2\kappa _{ij}+2\ell _{ij} \end{aligned}$$

and

$$\begin{aligned}&3m-h_{ij}=\mathcal {N}\left\{ (i,j):(x^{3bi}_{ik},x^{3bi}_{jk})\in \left\{ \left( \frac{1}{4},\frac{1}{4}\right) ,\left( \frac{3}{4},\frac{3}{4}\right) \right\} \right\} \\&\quad =3\beta _{ij}+3\alpha _{ij}+\gamma _{ij}+\rho _{ij}+\kappa _{ij}+\ell _{ij}. \end{aligned}$$

Thus, from the analytical expression $[{\mathcal {VD}^*}(\mathcal {U}_{3bi})]^2$ we get

$$\begin{aligned}{}[{\mathcal {VD}}(\mathcal {U}_{3bi})]^2&=\Delta (3m)+\frac{1}{n^{2}}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}(ab)^{3m-h_{ij}}a^{h_{ij}}\right) \\&=\Delta (3m)+\frac{1}{n^2}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}(ab)^{3\beta _{ij}+3\alpha _{ij}+\gamma _{ij}+\rho _{ij}+\kappa _{ij}+\ell _{ij}}a^{2\gamma _{ij}+2\rho _{ij}+2\kappa _{ij}+2\ell _{ij}}\right) \nonumber \\&=\Delta (3m)+\frac{1}{n^2}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}\left( {a^3}{b^3}\right) ^{\beta _{ij}+\alpha _{ij}}\left( {a^3}{b}\right) ^{\rho _{ij}+\kappa _{ij}+\ell _{ij}+\gamma _{ij}}\right) \nonumber \\&=\Delta (3m)+\frac{1}{n^2}{a^{3m}}{b^m}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}(b^2)^{\beta _{ij}+\alpha _{ij}}\right) . \end{aligned}$$

$\square$

Proof of Theorem 6

For any three binary Gray map image design $\mathcal {U}_{3bi}\in \mathbb {U}(n;2^{3m}),$ let the Hamming distance $h_{ij}=\mathcal {N}\left\{ (i,j):(x_{ik},x_{jk})\in \left\{ \left( \frac{1}{4},\frac{3}{4}\right) ,\left( \frac{3}{4},\frac{1}{4}\right) \right\} \right\} .$ Then, we have

$$\begin{aligned} h_{ij}=0 \quad \text{ and } \quad \sum _{i=1}^{n}\sum _{j\ne i}^{n}{h_{ij}}=\frac{3}{2}nm(n-2). \end{aligned}$$

(7.8)

The lower bound ${\mathcal {LBV}^*_{3bi}}$ is from the analytical expression $[{\mathcal {VD}^{*}}(\mathcal {U}_{3bi})]^2$ in Theorem 5, the relation (7.8) and Lemma 4 in Elsawah and Qin (2015c). The proof of the lower bound ${\mathcal {LBV}^{**}_{3bi}}$ is from the new analytical expression $[{\mathcal {VD}^{**}}(\mathcal {U}_{3bi})]^2$ in Theorem 5, the proof of Lemma 5 in Chatterjee et al. (2017) and Lemma 2 in Chatterjee et al. (2012) with some algebra, where we extend the existing results from two discrepancies, the $\mathcal {CD}$ and $\mathcal {WD},$ to all the five discrepancies, the $\mathcal {LD},~\mathcal {WD},~\mathcal {SD},~\mathcal {CD}$ and $\mathcal {MD},$ as well as we express the lower bounds of all these discrepancies as only one analytical expression $[{\mathcal {LBV}^{**}}(\mathcal {U}_{3bi})]^2.$ However, the proof of the lower bound ${\mathcal {LBV}^{***}_{2bi}}$ is given as follows. From the relations (a) and (c) in the proof of Theorem 2, we can rewrite the analytical formulas of different discrepancies in $[{\mathcal {VD}^{***}}(\mathcal {U}_{3bi})]^2$ as follows

$$\begin{aligned}{}[{\mathcal {VD}^{***}}(\mathcal {U}_{3bi})]^2= & {} \Delta (3m)+\frac{1}{n^2}{a^{3m}}{b^m}\left( \sum _{i=1}^{n}\sum _{j=1}^{n}(b^2)^{\beta _{ij}+\alpha _{ij}}\right) \nonumber \\= & {} \Delta (3m)+\frac{1}{n}{{a}^{3m}{b}^{3m}}+\frac{1}{n^2}{a^{3m}}{b^m}\left( \sum _{i=1}^{n}\sum _{j\ne i}^{n}(b^2)^{\beta _{ij}+\alpha _{ij}}\right) .\nonumber \\ \end{aligned}$$

(7.9)

The proof now is obvious from (7.9) and (7.7) by using Lemma 4 in Elsawah and Qin (2015c). $\square$