3-partition of a special set $S_5$ is a set consisting of the following 5-length sequences $s$: (1) each digit of $s$ is $a$, $b$, or $c$; (2) $s$ has and only has one digit that is $c$.
$T_5$ is a set consisting of the following 5-length sequences $t$: (1) each digit of $t$ is $a$, $b$, or $c$; (2) $t$ has two digits that are $c$.
Is there a 3-partition $S_5= \bigcup_{j=0,1,2} A_j$, $A_j 
\cap A_i=\varnothing$ with the following property: for any $A_j$ and any $t$ $\in$ $T_5$, there is a $s$ $\in$ $A_j$ such that exsits a $n\in\{1,2,3,4,5\}$, $s_n \neq t_n$, $t_n=c$ and $s_m=t_m $ for any $m\neq n$, where $s_n$ (or $t_n$) is the n-th digit of $s$ (or $t$).
For example, $t=ccaab$ and $s=acaab$ where $n=1$.
And is there a 3-partition of $S_6$ with $T_6$ for 6-length sequences?
Thank you!
 A: Here is one such partition of $S_5$, obtained via integer linear programming:
\begin{align}
A_0 &= \{aaaac,aaacb,aabca,aacba,ababc,abbac,abcab,abcba,acaaa,acbbb,baabc,
babac,bacaa,bacbb,bbaca,bbbbc,bbbcb,bbcaa,bcaab,bcbba,caaba,cabab,cabbb,
cbaab,cbabb,cbbaa\} \\
A_1 &= \{aaaca,aabbc,aacab,abaac,abbca,abbcb,abcbb,acaba,acabb,acbaa,acbab,
baaac,babca,babcb,bacba,bbabc,bbacb,bbbac,bbcab,bbcba,bcaaa,bcbbb,caaab,
caabb,cabaa,cbaaa,cbbba\} \\
A_2 &= \{aaabc,aabac,aabcb,aacaa,aacbb,abaca,abacb,abbbc,abcaa,acaab,acbba,
baaca,baacb,babbc,bacab,bbaac,bbbca,bbcbb,bcaba,bcabb,bcbaa,bcbab,caaaa,
cabba,cbaba,cbbab,cbbbb\}
\end{align}
And here is one for $S_6$:
\begin{align}
A_0 &= \{aaaaac,aaabca,aaacbb,aababc,aabacb,aabbac,aabcba,aacaaa,aacbab,aacbba,
abaaac,abaacb,ababbc,abacaa,abbabc,abbbca,abbcab,abcaba,abcbaa,abcbbb,acaaba
,acaabb,acabab,acbaaa,acbaab,acbbbb,baaabc,baabcb,baacaa,babaac,babaca,
babbbc,babcab,bacabb,bacbab,bacbba,bbaabc,bbaaca,bbabac,bbacbb,bbbaac,bbbbcb
,bbbcba,bbcaab,bbcbaa,bbcbbb,bcaaaa,bcaaab,bcabba,bcbaba,bcbabb,bcbbaa,
caaaab,caaaba,caabaa,caabbb,cabbaa,cabbbb,cbabab,cbabba,cbbaaa,cbbabb,cbbbab
,cbbbba\} \\
A_1 &= \{aaaaca,aaabbc,aaacab,aabaac,aabbbc,aabbcb,aabcaa,aacaba,aacabb,aacbaa,
abaabc,ababac,ababcb,abacba,abbaca,abbbac,abbcbb,abcaaa,abcaab,abcbba,acaaab
,acabaa,acabbb,acbabb,acbbab,acbbba,baaacb,baabac,baacba,bababc,babbac,
babbca,babcbb,bacaaa,bacaab,bacbbb,bbaaac,bbabbc,bbabca,bbacab,bbbacb,bbbbbc
,bbbcaa,bbcaba,bbcabb,bbcbab,bcaaba,bcabaa,bcabbb,bcbaaa,bcbbab,bcbbba,
caaaaa,caaabb,caabab,caabba,cabaab,cababa,cbaaaa,cbaabb,cbbaab,cbbaba,cbbbaa
,cbbbbb\} \\
A_2 &= \{aaaabc,aaaacb,aaabac,aaabcb,aaacaa,aaacba,aabaca,aabbca,aabcab,aabcbb,
aacaab,aacbbb,abaaca,ababca,abacab,abacbb,abbaac,abbacb,abbbbc,abbbcb,abbcaa
,abbcba,abcabb,abcbab,acaaaa,acabba,acbaba,acbbaa,baaaac,baaaca,baabbc,
baabca,baacab,baacbb,babacb,babbcb,babcaa,babcba,bacaba,bacbaa,bbaacb,bbabcb
,bbacaa,bbacba,bbbabc,bbbaca,bbbbac,bbbbca,bbbcab,bbbcbb,bbcaaa,bbcbba,
bcaabb,bcabab,bcbaab,bcbbbb,cabaaa,cababb,cabbab,cabbba,cbaaab,cbaaba,cbabaa
,cbabbb\}
\end{align}
