Character to bits with SIMD (and substrings) - sse

Question

I am learning little by little SIMD programming, and I've devised a (seemingly) simple problem that I hope I can speed-up using SIMD (AVX, at the moment I have access only to AVX CPUs).
I have a long string constituted by an alphabet of 2^k characters (for instance 0, 1, 2, 3), and I'd like to:
generate all substrings of a given length substringlength
convert all the substrings in bits
The substrings are just sequences of characters from the input string:
012301230123012301230123012301233012301301230123123213012301230
substringlength = 6;
string bits
------+--+-----------------
012301 -> 01 00 11 10 01 00
123012 -> 10 01 00 11 10 01
230123 -> 11 10 01 00 11 10
301230 -> 00 11 10 01 00 11
...
My question is due to my inexperience with SIMD (I've only read "Modern x86 Assembly Language Programming", by Kusswurm):
Is this a task where SIMD could help?
Edit: for simplicity, let's just assume k = 2, and so the ASCII numbers will be just '0'..'3'.
Iteration 1
Reading the comments and playing around I've come to these realizations. I can convert the the ASCII into values, and as suggested, multiply-add adjacent bytes:
// SIMD 128-bit registers, apparently I cannot use AVX ones directly (some operations are AVX2 or AVX-512)
__m128i sse, val, adj, res;
auto mask = _mm_set_epi8(1, 1<<4, 1, 1<<4, 1, 1<<4, 1, 1<<4, 1, 1<<4, 1, 1<<4, 1, 1<<4, 1, 1<<4);
auto zero = _mm_set_epi8('0', '0', '0', '0', '0', '0', '0', '0',
'0', '0', '0', '0', '0', '0', '0', '0');
// Load ascii values
sse = _mm_loadu_si128((__m128i*) s.data());
// Convert to integer values
val = _mm_sub_epi8(sse[0], zero);
// Multiply with mask byte by byte (aka SHL second bytes of val) and sum
adj = _mm_maddubs_epi16(val, mask);
An idea of what it does, to people learning like me, is given here (I will need more 128-bits to encode one substring, ascii is in hex):
bytes 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
ascii 30 31 30 31 30 31 30 31 30 31 30 31 30 31 30 31
_mm_sub_epi8:
value 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
_mm_maddubs_epi16:
value 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
* * * * * * * * * * * * * * * *
mask 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4
+ + + + + + + +
| | | | | | | | |
(16-bits)
bits ....0100 ....0100 ....0100 ....0100 ....0100 ....0100 ....0100 ....0100
In other words the first 4 bits are correct, encoding 2 ascii chars, if I understand correctly what _mm_maddubs_epi16 did to my values, which I am not sure at all!
Now I'd need some sort of "shift-or" of adjacent bytes, something like _mm_maddubs_epi16 that shifts left the first, and ORs with the second argument, producing an 8-bit or 16-bit value:
(16-bits)
bits ....0100 ....0100 ....0100 ....0100 ....0100 ....0100 ....0100 ....0100
| shl 4 | | shl 4 | | shl 4 | | shl 4 |
0100.... ....0100 0100.... ....0100 0100.... ....0100 0100.... ....0100
OR OR OR OR
....01000100 ....01000100 ....01000100 ....01000100
However, I cannot see how _mm_bslli_si128 could help me here, or if there is a smarter way to do this. Maybe even this "horizontal" approach is foolish, and I have to rethink it.
Any hint is welcome!