A letter-by-letter Hamming code for manual ECC computation

Watched 3Blue1Brown’s video on Hamming codes recently and a couple of thoughts occurred to me.

First, Hamming codes, like matrix parity codes, are simple enough that you could reasonably compute them by hand, making them a reasonable candidate for archival media.

Second, you can take the same Hamming-code approach over any character code, not just a binary code. For example, rather than computing a (15, 11) Hamming code by adding 4 parity bits to 11 data bits, or a (7, 4) Hamming code by adding 3 parity bits to 4 data bits, you could add 4 “parity” letters to 11 data letters, or 3 “parity” letters to 4 data letters, or indeed 6 “parity” letters to 57 data letters; a variety of “parity” computations are possible but perhaps the simplest is to use a character code assigning numbers 0 to n-1 to the possible letters, and use the sum modulo n. (It’s entirely irrelevant what n is, but the decoder needs to know the whole code.) This is optimized for situations in which a whole letter at a time is damaged or lost, rather than single-bit errors.

Third, you can run either variant of the Hamming-code approach along various axes. If your text consists of lines of up to 57 characters, for example, you could add 6 parity characters (or 7, for a SECDED extended Hamming code) to each line, or you could divide it into “pages” of 57 lines and add 6 or 7 parity lines, each of whose characters would be computed over the corresponding characters in the other lines. This would enable the recovery of entire missing or erroneous lines.

Fourth, you can combine this with the matrix-parity idea; for example, you could compute an extended Hamming code both horizontally and vertically, allowing you to correct up to one error per line, plus up to one line with two or more errors. This is not the most efficient error-correcting code, but it is very simple, and enables a substantial level of robustness.

If you were using this for archival in practice, you might want to put the “parity” lines and columns at the beginning or end of the data, rather than interspersing them as in the canonical Hamming-code construction.

The ASCII character code has some disadvantages as a code to use in this context, since its last position is an unprintable character (DEL) and so are its first 32 positions, except arguably TAB, CR, LF, and BEL. Also, arguably, space is unprintable; certainly it is especially prone to OCR errors. But if you replace the unprintable characters with printable ones — one option would be “␀␁␂␃␄␅␆␇␈␉␊␋␌␍␎␏␐␑␒␓␔␕␖␗␘␙␚␛␜␝␞␟␣␥”, but many others have been used at different times, including accented letters and extra punctuation — then you would have an error-correction code people could very plausibly discover by hand in an archival document, for example if microprinted.

TeX OT1

One particularly handy 7-bit all-printable encoding for text is TeX’s OT1 font encoding, whose translation into Unicode Wikipedia gives as follows, if I haven't screwed it up:

Γ U+0393
Δ U+0394
Θ U+0398
Λ U+039B
Ξ U+039E
Π U+03A0
Σ U+03A3
Υ U+03A5
Φ U+03A6
Ψ U+03A8
Ω U+03A9
ff U+FB00
fi U+FB01
fl U+FB02
ffi U+FB03
ffl U+FB04
ı U+0131
ȷ U+0237
` U+0060
´ U+00B4
ˇ U+02C7
˘ U+02D8
ˉ U+02C9
˚ U+02DA
¸ U+00B8
ß U+00DF
æ U+00E6
œ U+0153
ø U+00F8
Æ U+00C6
ΠU+0152
Ø U+00D8
̷ U+0337
! U+0021
” U+201D
# U+0023
$ U+0024
% U+0025
& U+0026
’ U+2019
( U+0028
) U+0029
* U+002A
+ U+002B
, U+002C
- U+002D
. U+002E
/ U+002F
0 U+0030
1 U+0031
2 U+0032
3 U+0033
4 U+0034
5 U+0035
6 U+0036
7 U+0037
8 U+0038
9 U+0039
: U+003A
; U+003B
¡ U+00A1
= U+003D
¿ U+00BF
? U+003F
@ U+0040
A U+0041
B U+0042
C U+0043
D U+0044
E U+0045
F U+0046
G U+0047
H U+0048
I U+0049
J U+004A
K U+004B
L U+004C
M U+004D
N U+004E
O U+004F
P U+0050
Q U+0051
R U+0052
S U+0053
T U+0054
U U+0055
V U+0056
W U+0057
X U+0058
Y U+0059
Z U+005A
[ U+005B
“ U+201C
] U+005D
ˆ U+02C6
˙ U+02D9
‘ U+2018
a U+0061
b U+0062
c U+0063
d U+0064
e U+0065
f U+0066
g U+0067
h U+0068
i U+0069
j U+006A
k U+006B
l U+006C
m U+006D
n U+006E
o U+006F
p U+0070
q U+0071
r U+0072
s U+0073
t U+0074
u U+0075
v U+0076
w U+0077
x U+0078
y U+0079
z U+007A
– U+2013
— U+2014
˝ U+02DD
˜ U+02DC
¨ U+00A8

This is intentionally missing some mathematical symbols, namely <, >, {, |, and }, as well as some others rarely used in text like \\ and _. TeX generally sets mathematical symbols in a different font (which makes it a bit strange to include some Greek letters but not enough to actually write Greek), and can stack glyphs one on top of the other, so separate codepoints for letters like áéòč are not needed.