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J, 6 distinct chars

6 distinct char version (u:10 +) using some idea flornquake used. The method is adding vectors of 0 1 11 and 111 together to add up to the desired 84 104 101 32 113 117 ... vector:

 u:11 11 11 11 111 111 11 11 11 11 11 111 111 111 11 11 11 111 111 11 11 111 11 111 111 11 111 111 11 111 11 111 11 11 11 11 11 111 111 11 11 111 11 + 11 11 11 11 1 1 11 11 11 11 11 1 0 1 11 11 11 0 1 11 11 1 11 1 1 11 0 1 11 1 11 1 11 11 11 11 11 11 1 11 11 0 11 + 11 11 11 1 1 1 11 11 11 1 11 1 0 1 11 1 11 0 1 1 11 1 11 0 1 1 0 1 11 1 1 1 11 11 1 11 11 0 1 1 11 0 11 + 11 11 11 1 0 1 11 11 11 1 11 1 0 1 11 1 11 0 1 1 11 1 11 0 1 1 0 1 11 1 1 1 11 11 1 11 11 0 1 1 11 0 11 + 11 11 11 1 0 1 11 11 11 1 11 0 0 1 11 1 11 0 1 1 11 1 11 0 1 1 0 1 11 0 1 1 11 11 1 11 11 0 1 1 11 0 11 + 11 11 11 1 0 1 11 11 11 1 11 0 0 1 11 1 11 0 1 1 11 1 11 0 0 1 0 1 11 0 1 1 11 11 1 11 11 0 1 1 11 0 11 + 11 11 11 1 0 1 11 11 11 1 11 0 0 1 11 1 11 0 1 1 11 1 11 0 0 1 0 1 11 0 1 0 11 11 1 11 11 0 1 1 11 0 11 + 1 11 11 1 0 0 11 11 11 1 11 0 0 1 11 1 11 0 1 1 11 0 11 0 0 1 0 1 11 0 1 0 11 11 1 11 11 0 1 1 11 0 11 + 1 11 11 1 0 0 11 11 11 1 1 0 0 1 11 1 11 0 1 1 11 0 11 0 0 1 0 0 11 0 1 0 11 11 1 11 1 0 1 1 11 0 11 + 1 1 1 1 0 0 1 0 1 1 1 0 0 0 11 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 1 0 1 + 1 1 1 1 0 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 1 + 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 + 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 + 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 + 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The quick brown fox jumps over the lazy dog 

13 distinct char version:

 u: 84 104 101 32 113 117 105 99 107 32 98 114 111 119 110 32 102 111 120 32 106 117 109 112 115 32 111 118 101 114 32 116 104 101 32 108 97 122 121 32 100 111 103 The quick brown fox jumps over the lazy dog 

Anonymous improvement: shorter solution but still using 6 characters (u:10 |) is possible. Using | as residue operator for decimals e.g. 1011 | 1100110001100 = 84.

 u:1011 1101 1011 1011 1111 1011 1101 1001 1101 1011 1001 1101 1011 1001 1011 1011 1101 1011 1111 1011 1101 1011 1001 1111 1101 1011 1011 1001 1011 1101 1011 1011 1101 1011 1011 1001 1101 1111 1111 1011 1011 1011 1111|1100110001100 10101010100 101 1100000 100010111 110001110010 10101010101 1100 100010000111 1100000 1000001100 10101010110 111 10001110 110 1100000 10011 111 100110 1100000 100010000110 110001110010 1110 10111 10101010111 1100000 111 1010001110 101 10101010110 1100000 1100010001100 10101010100 101 1100000 1000001110 100000101010 110111 100111 1100000 100 111 100010101 The quick brown fox jumps over the lazy dog 

And that solution can further be improved using (u:23 |). (DirkL)

u:223|2333333 232333323 3223 32 23223333 23222222 323232 322 32333323 32 322333 232333333 3233 2323333 3232 32 2332 3233 32232 32 323233 23222222 2323323 23223332 3223333322 32 3233 2323332 3223 232333333 32 32333332 232333323 3223 32 2323322 322332 32332223 32233 32 323 3233 2333

The quick brown fox jumps over the lazy dog

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