An Introduction to
Binary Chain Codes

SUMMARY:   A Don Lancaster tutorial and Jim Fitzsimons math wonderment that
                         explores binary chain codes. These are unique sequences where
                         you can determine your position in the sequence in very few bytes.

                         Includes BASIC program and listing of possible and verified
                         chain codes for order 2 through 16.

                        May be read as tutorial with any word processor or editor.
                        Actual use requires BASIC language.

                       Copyright c 1994 by Jim Fitzsimons. All rights reserved.
                       Free help line and additional info: (520) 428-4073.

Name of textfile:        BINCHAIN.TXT
Source:                       J.FITZSIMONS
Authors:                     Don Lancaster & Jim Fitzsimons
Desc:                          Binary chain code introduction
Date:                          May 5, 1994 Revised Oct 1 1996
Release:                    1.0

Status:                       Copyright 1994 by J.Fitzsimons
                                  All commercial rights and all electronic media rights
                                  fully reserved. Reposting is expressly forbidden.
                                  Royalties and consulting fees required for any and all
                                  commercial uses.

Additional info:        (602) 428-4073 or www.tinaja.com.

Keywords:                Binary, chain, code, math, sequence, position, code,
                                  encoder, position, shaft, utility, tutorial, one, zero

Binary Chain Sequences
By Don Lancaster & Jim Fitzsimons

A recent issue of SENSORS magazine had a stunningly simple absolute shaft
encoder in it. I (don) still don't understand the full details, but it was
clear that a binary chain code was involved. We are still trying to find
more details on this.

At any rate, we decided that a binary chain code was a sequence of ones and
zeros of order n. This means that n bits are involved and the serial sequence
is 2^n long. The sequence wraps around itself, so there is no start or finish.

The sequence is designed so that ANY sample of n bits will tell you your
EXACT position in the code string! All strings of n bits are thus unique
and correspond to all possible values of length n.

The real chain codes are much fancier than this, so stay tuned. Besides
greatly simplified shaft encoders, they apparently have use in graphical
compression of engineering documents, and precise positioning aps.

Here's an example. One possible 3 bit chain code is $17. Write this
out in binary....

0 0 0 1 0 1 1 1

Then add a few bits, chaining things end to end...

0 0 0 1 0 1 1 1 0 0 0 1 ....

0 0 0
0 0 1
0 1 0
1 0 1
0 1 1
1 1 1
1 1 0
1 0 0
0 0 0 (repeats...)

Note that each position has a unique code and that each possible
3-bit value is represented only once. Thus by knowing only n bits
you know exactly where you are in the entire 2^n series.

Other properties of chain codes are that they have identical numbers
of ones and zeros, and never more than n ones or zeros in sequence.
Thus, a valid one can always start with n zeros in a row, followed by
at least one one.

Although there are mirror and complement and phase shifted versons,
chain codes are quite rare and hard to find for higher n. Jim's
algorithm seem to work like a champ. The maximum possible number of
chain codes is upper bounded by the bionomial expansion coefficient for
n/2 out of n.

This obviously could be converted to PostScript. But this is only
step one in finding out more about chain codes.

Here is how to generate your own codes of order 2 through 16, plus
a printout of them...

Jim's Chain Code Generator:

REM CHAIN.BAS JIM 05APR94 07:07
OPEN "CHAIN" FOR OUTPUT AS #1
OPEN "CHAINB" FOR OUTPUT AS #2
INPUT "limit"; LP%
TN& = 2 ^ LP%
TNB% = 16: TNA% = TN& / TNB%
REDIM H&(TNB% - 1, TNA% - 1), L%(TNB% - 1, TNA% - 1), F%(TNB% - 1, TNA% - 1)
FOR P% = 4 TO LP%
TN& = 2 ^ P% - 1
CLS
FOR I& = 0 TO TN&
IA% = I& \ TNA%: IB% = I& MOD TNA%
H&(IA%, IB%) = 0
L%(IA%, IB%) = 0
F%(IA%, IB%) = 0
NEXT I&
N& = 0
F%(0, 0) = -1
DO
I& = N& + 1
IA% = N& \ TNA%: IB% = N& MOD TNA%
A& = H&(IA%, IB%)
DO
A& = A& * 2 OR L%(IA%, IB%)
B& = A& AND TN&
IA% = B& \ TNA%: IB% = B& MOD TNA%
IF F%(IA%, IB%) THEN
EXIT DO
END IF
F%(IA%, IB%) = -1
IA% = I& \ TNA%: IB% = I& MOD TNA%
H&(IA%, IB%) = B&
N& = I&
I& = I& + 1
IF I& > TN& THEN EXIT DO
LOOP
IF I& > TN& THEN EXIT DO
DO
IF N& < 0 THEN END
IA% = N& \ TNA%: IB% = N& MOD TNA%
IF L%(IA%, IB%) = 0 THEN
L%(IA%, IB%) = 1
EXIT DO
END IF
PRINT N&;
L%(IA%, IB%) = 0
I& = H&(IA%, IB%)
IA% = I& \ TNA%: IB% = I& MOD TNA%
F%(IA%, IB%) = 0
N& = N& - 1
LOOP
K$ = INKEY$
LOOP UNTIL K$ = CHR$(27)
T% = 0
M& = (TN& + 1) \ 2
J% = 0
PRINT #2, P%
FOR I& = 0 TO TN&
IA% = I& \ TNA%: IB% = I& MOD TNA%
IF (H&(IA%, IB%) AND M&) <> 0 THEN
T% = 1 + T% * 2
PRINT #2, "1";
ELSE
T% = T% * 2
PRINT #2, "0";
END IF
IF (I& MOD 80) = 79 THEN PRINT #2, ""
J% = J% + 1
IF J% = 4 THEN
PRINT HEX$(T%);
PRINT #1, HEX$(T%);
J% = 0
T% = 0
IF (I& MOD 320) = 319 THEN PRINT #1, ""
END IF
NEXT I&
PRINT #1, ""
PRINT #2, ""
IF K$ = CHR$(27) THEN EXIT FOR
NEXT P%
CLOSE #1
CLOSE #2
END

2 bit chain code:

$63 bit chain code:

$174 bit chain code:

09AF5 bit chain code:

04653ADF6 bit chain code:

0218A392CD3D5DBF 7 bit chain code:

01061438916347932A5CD9D3EAD7B77F8 bit chain code:

00818283848586878898A8B8C8D8E8F929395969799A9B9D9E9FAAEB6BEDEEFF9 bit chain code:

004060503824160D0784426150B864361D0F88C4A27148AC5A2F198D46E391D8 F47E4964D2794CAA572D974BE67359B4DE753B9ECFA7F55ABD6EBB5FB6FBBDFF10 bit chain code:

0020180A038120581A078220982A0B8320D83A0F8423094270A42B0B42F0C433 0D4370E43B0F43F114471244B1344F14C5515C5916C5D17C6328CE348D6368DE 398EA3B8F23D8FA3F9264A92E4D93A4F94A72ACAD2BCB32D4B72ECBD2FCCD33C D535CDB374DF39D6769DE7A9EE7D9FA7FAABAB6AFAD6F5DD7B5FDB76FDDF7BFF11 bit chain code:

00100601403809016034078110260540B8190360740F821046094138290560B4 178310660D41B8390760F41F84308A11C2484B09A13C288530AA15C2C85B0BA1 7C318650CE1A43586D0DE1C4398750EE1E43D87D0FE2244588D11E24C4A89713 226C4E89F146294538A91562B4578B316A2DC5C8BB17A2FC638C9196334678D3 1AA35C6C8DB1BA37C728E71D23AC768EF1E63D47B8F91F63F47F92524E4AC969 2F2664D49B93B27A4FCA594D29E54CAA9572B6574AF9672D65B4B79732EA5DCB D97D2FE66CCE99F34E6ACD69AF36A6DCDD9BD37E73CEA9D73B6774EF9EB3DA7B CFA9F73F67F4FFAAB55EADD5DABF5AEB6D6FAEF5EEBED7FB6F6EEDFDDFBDF7FF12 bit chain code:

00080180280380480580680780880980A80B80C80D80E80F8108118128138148 1581681781881981A81B81C81D81E81F8208608A08E09209609A09E0A20A60AA 0AE0B20B60BA0BE0C20C60CA0CE0D20D60DA0DE0E20E60EA0EE0F20F60FA0FE1 0A10E11211611A11E12212612A12E13213613A13E14614A14E15215615A15E16 216616A16E17217617A17E186286386486586686786886986A86B86C86D86E86 F87187287387487587687787887987A87B87C87D87E87F888988A88B88C88D88 E88F89189289389489589689789989A89B89C89D89E89F8A28E29229629A29E2 A62AA2AE2B22B62BA2BE2C62CA2CE2D22D62DA2DE2E62EA2EE2F22F62FA2FE31 A31E32632A32E33233633A33E34A34E35235635A35E36636A36E37237637A37E 38E48E58E68E78E98EA8EB8EC8ED8EE8EF8F28F38F48F58F68F78F98FA8FB8FC 8FD8FE8FF924B24D24F25325525725B25D25F26526726B26D26F27327527727B 27D27F29529729B29D29F2A72AB2AD2AF2B32B52B72BB2BD2BF2CB34B3CB4CB5 4B5CB6CB74B7CB9CBACBB4BBCBCCBD4BDCBECBF4BFCCCD4CDCCECCF4CFCD34F3 5535735B35D35F36736B36D36F37537737B37D37F39D39F3AB3AD3AF3B53B73B B3BD3BF3CF54F5CF6CF74F7CFACFB4FBCFD4FDCFECFF4FFD55755B55F56B56F5 7757B57F5AF5B75BB5BF5D76D77D7BD7DD7ED7FDB6FB77B7BB7FBBBFBDFBEFFF13 bit chain code:

0004006005003802401600D007804402601500B806403601D00F808404602501 380A405602D01780C406603501B80E407603D01F8104086045023812409604D0 2781440A605502B81640B605D02F81840C606503381A40D606D03781C40E6075 03B81E40F607D03F820C10A087044822C11A08F048824C12A09704C826C13A09 F050828C14A0A705482AC15A0AF05882CC16A0B705C82EC17A0BF061831418E0 C9065833419E0D106983541AE0D906D83741BE0E107183941CE0E907583B41DE 0F107983D41EE0F907D83F41FE109085843421E111089845422E11908D847423 E123092849C25212B09684BC26213309A84DC27213B09E84FC2861450A385242 9614D0A785442A61550AB85642B615D0AF858C2CA1670B485AC2DA16F0B885CC 2EA1770BC85EC2FA17F0C3862431618D0C786443261950CB866433619D0CF868 C34A1A70D486AC35A1AF0D886CC36A1B70DC86EC37A1BF0E2871C3921CB0E687 3C3A21D30EA875C3B21DB0EE877C3C61E50F387A43D61ED0F787C43E61F50FB8 7E43F61FD0FF888C44A22711488AC45A22F11988D446E23911D88F447E245123 892449624D127894C4AA25712C896C4BA25F13189944CE26913589B44DE27313 A89DC4F227B13E89FC51628D1478A4C52A29714C8A6C53A29F1518A9454E2A91 558AB455E2B315A8ADC5722BB15E8AFC58E2C91658B3459E2D316A8B5C5B22DB 16E8B7C5C62E51738BA45D62ED1778BCC5EA2F717C8BEC5FA2FF18C8C6C63A31 F1928C9C65232B1968CBC66633519B8CE467633D19F8D1C69234B1A68D3C6A63 551AB8D646B635D1AF8D946CE3691B58DB46DE3731BA8DDC6F237B1BE8DFC71E 3931CA8E5C73239B1CE8E7C74A3A71D48EAC75A3AF1D98ED476E3B91DD8EF477 E3C91E58F3479E3D31EA8F5C7B23DB1EE8F7C7CA3E71F48FAC7DA3EF1F98FD47 EE3F91FD8FF47FE49324A925C93649D24F929494E4AB25692BC9664B525B92EC 97A4BF265933499E4D326A935C9B64DD26F93949CE4EB27693BC9E64F527B93E C9FA4FF29594B4A5E53329A94DCA7653D29F952CA9A54F2A99554AAE55B2AE95 7CACE56B2B695BCAE65752BB95ECAFA57F2CB966CB3A59F2D396ACB5A5AF2D99 6D4B6E5BB2DE96FCB9A5CF2E99754BAE5DB2EE977CBCE5EB2F697BCBE65F52FB 97ECBFA5FF33399ACCDA66F33A99DCCF667D33F9A6CD3A69F3539AACD5A6AF35 A9ADCD766BD35F9B3CDAA6D736D9B74DBE6E73759BB4DDE6F537B9BECDFA6FF3 9D9CF4E7E74F3AA9D5CEB675D3AF9DACEDA76F3BA9DDCEF677D3BF9E7CF567AD 3D79ED4F6E7BB3DE9EFCFAA7D73ED9F74FBE7EB3F69FBCFEA7F73FD9FF4FFEAA B557AADD576ABF56BAB6D5BEAEB577ABDD5F6AFF5ADAD7D6DEB775BDADFD75EB B75DDAEFD7B6BDF5F7AFDD7F6BFF6DBB6FDBBEDEF6FBB7FDDDEEFF7BFBEFDFFF14 bit chain code:

0002001800A0038012005801A0078022009802A00B803200D803A00F80420118 04A0138052015805A0178062019806A01B807201D807A01F8082021808A02380 92025809A02780A202980AA02B80B202D80BA02F80C203180CA03380D203580D A03780E203980EA03B80F203D80FA03F81020C085021C089022C08D023C09102 4C095025C099026C09D027C0A1028C0A5029C0A902AC0AD02BC0B102CC0B502D C0B902EC0BD02FC0C1030C0C5031C0C9032C0CD033C0D1034C0D5035C0D9036C 0DD037C0E1038C0E5039C0E903AC0ED03BC0F103CC0F503DC0F903EC0FD03FC1 05041C109042C10D043C111044C115045C119046C11D047C121048C125049C12 904AC12D04BC13104CC13504DC13904EC13D04FC1430514147052414B053414F 05441530554157056415B057415F0584163059416705A416B05B416F05C41730 5D417705E417B05F417F060C2830E0C483160C6831E0C883260CA832E0CC8336 0CE833E0D083460D2834E0D483560D6835E0D883660DA836E0DC83760DE837E0 E1838A0E383920E5839A0E783A20E983AA0EB83B20ED83BA0EF83C20F183CA0F 383D20F583DA0F783E20F983EA0FB83F20FD83FA0FF8422109842A10B843210D 843A10F8446112844E1148456116845E118846611A846E11C847611E847E1218 48A1238492125849A12784A212984AA12B84B212D84BA12F84C613284CE13484 D613684DE13884E613A84EE13C84F613E84FE142870A2428B0A3428F0A442930 A542970A6429B0A7429F0A8C2A50A9C2A90AAC2AD0ABC2B10ACC2B50ADC2B90A EC2BD0AFC2C30B142C70B242CB0B342CF0B442D30B542D70B642DB0B742DF0B8 C2E50B9C2E90BAC2ED0BBC2F10BCC2F50BDC2F90BEC2FD0BFC30D0C3C3110C4C 3150C5C3190C6C31D0C7C3230C943270CA432B0CB432F0CC43330CD43370CE43 3B0CF433F0D143470D2434B0D3434F0D443530D543570D6435B0D7435F0D8C36 50D9C3690DAC36D0DBC3710DCC3750DDC3790DEC37D0DFC387121C5871A1C787 221C9872A1CB87321CD873A1CF87461D2874E1D487561D6875E1D887661DA876 E1DC87761DE877E1E2878E1E487961E6879E1E887A61EA87AE1EC87B61EE87BE 1F187CA1F387D21F587DA1F787E21F987EA1FB87F21FD87FA1FF888A22388922 25889A22788A622A88AE22C88B622E88BE23188CA23388D223588DA23788E623 A88EE23C88F623E88FE2448B123448F124C495125C499126C49D127C4A312944 A712A44AB12B44AF12CC4B512DC4B912EC4BD12FC4C5131C4C9132C4CD133C4D 313544D713644DB13744DF138C4E5139C4E913AC4ED13BC4F313D44F713E44FB 13F44FF1454517146451B147451F148C525149C52914AC52D14BC53314D45371 4E453B14F453F151C549152C54D153C5531554557156455B157455F158C56515 9C56915AC56D15BC57315D457715E457B15F457F162C68B1E2C98B2A2CB8B322 CD8B3A2CF8B462D28B4E2D48B562D68B5E2D98B6A2DB8B722DD8B7A2DF8B8E2E 48B962E68B9E2E98BAA2EB8BB22ED8BBA2EF8BC62F28BCE2F48BD62F68BDE2F9 8BEA2FB8BF22FD8BFA2FF8C6631A8C6E31C8C7631E8C7E3238C923258C9A3278 CA632A8CAE32C8CB632E8CBE3328CCE3348CD63368CDE3398CEA33B8CF233D8C FA33F8D1A3C6931A546971A6469B1A7469F1A946A71AA46AB1AB46AF1ACC6B51 ADC6B91AEC6BD1AFC6C71B246CB1B346CF1B4C6D51B5C6D91B6C6DD1B7C6E51B 9C6E91BAC6ED1BBC6F31BD46F71BE46FB1BF46FF1C7471F1C947271CA472B1CB 472F1CCC7351CDC7391CEC73D1CFC7491D2C74D1D3C7531D547571D6475B1D74 75F1D947671DA476B1DB476F1DCC7751DDC7791DEC77D1DFC78F263CA8F2E3CC 8F363CE8F3E3D28F4E3D48F563D68F5E3D98F6A3DB8F723DD8F7A3DF8F923E58 F9A3E78FA63EA8FAE3EC8FB63EE8FBE3F28FCE3F48FD63F68FDE3F98FEA3FB8F F23FD8FFA3FF924A4939256496925E499926A49B927649E927E4A5929A4A792A 64AA92AE4AD92BA4AF92CA4B392D64B692DE4B992EA4BB92F64BE92FE4C99526 5C99B267499F26949A726AC9AD26BC9B326D49B726EC9BD26FC9CB27349CF274 C9D5275C9DB27749DF27949E727AC9ED27BC9F327D49F727EC9FD27FCA532954 A57296CA5D297CA6729ACA6D29BCA7329D4A7729ECA7D29FCA952E54D953A54F 954E555955A557956655A956E55D957A55F9596566959E56995AA56B95B656E9 5BE57395D657695DE57995EA57B95F657E95FE596965E599966A59B967659E96 7E5A6969E5A996AA5AB96B65AE96BE5B396D65B696DE5B996EA5BB96F65BE96F E5CB9B2E74B9F2E9CBAB2EB4BAF2ECCBB52EDCBBB2EF4BBF2F34BCF2F4CBD52F 5CBDB2F74BDF2F9CBEB2FB4BEF2FCCBF52FDCBFB2FF4BFF3334CCF3354CD7336 CCDD337CCE733ACCED33BCCF533DCCFB33F4CFF34D4D3734ECD3D34FCD4F3554 D57356CD5D357CD6735ACD6D35BCD7535DCD7B35F4D7F366CE9B3E6D39B566D6 9B5E6DA9B6E6DD9B7A6DF9B9E6EA9BAE6ED9BBA6EF9BCE6F59BDA6F79BEA6FB9 BF66FE9BFE73A9CEE73D9CFA73F9D3A7CEAB3AB4EAF3AD4EB73AECEBD3AFCECF 3B54ED73B6CEDD3B7CEEB3BB4EEF3BD4EF73BECEFD3BFCF3D3CFCF553D5CF5B3 D74F5F3DACF6D3DBCF753DDCF7B3DF4F7F3E7D59F5A7D79F6A7DB9F767DE9F7E 7EA9FAE7ED9FBA7EF9FD67F69FDE7FA9FEE7FD9FFA7FFAAABAAB6AAFAAD6AB7A AEEABDAAFEAD5AF56DD5BB56FD5D7576D5DF57AD5EF57DD5FB57FD6B75AED6BF 5B5D6DB5B7D6EF5BDD6FB5BFD75F5DBD7775DED77F5EBDBAF76BDFAFB6BEFAFD EBFBAFF6BFFB6DEDBBB6FEDDBBF6F7DBEF6FDDBFF777DDEF77FDEFF7DFDFBFFF15 bit chain code:

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