View source for Two-out-of-five code

{{Short description|Error-detection code for decimal digits}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
[[Image:Barcode2of5example.svg|thumb| 2 of 5 barcode (non-interleaved)]]
[[Image:Correios CEP 22011-080 - Codigo de Barras - PU1JFC.jpg|thumb| [[POSTNET]] barcode]]
A '''two-out-of-five code''' is a [[constant-weight code]] that provides exactly ten possible combinations of two bits, and is thus used for representing the [[decimal digit]]s using five [[bit]]s.<ref name="MIL_1991"/>  Each bit is assigned a weight, such that the set bits sum to the desired value, with an exception for zero.

According to [[Federal Standard 1037C]]:
* each decimal [[numerical digit|digit]] is represented by a binary numeral consisting of five bits of which two are of one kind, called ''ones'', and three are of the other kind, called ''zeros'', and
* the usual weights assigned to the bit positions are 0-1-2-3-6.  However, in this scheme, [[0|zero]] is encoded as binary ''01100''; strictly speaking the 0-1-2-3-6 previously claimed is just a [[mnemonic]] device.<ref>{{FS1037C}}</ref>

The weights give a unique encoding for most digits, but allow two encodings for 3: 0+3 or 10010 and 1+2 or 01100.  The former is used to encode the digit 3, and the latter is used to represent the otherwise unrepresentable zero.

The [[IBM 7070]], [[IBM 7072]], and [[IBM 7074]] computers used this code to represent each of the ten decimal digits in a machine word, although they numbered the bit positions 0-1-2-3-4, rather than with weights.  Each word also had a sign flag, encoded using a two-out-of-three code, that could be '''A''' Alphanumeric, '''−''' Minus, or '''+''' Plus.  When copied to a digit, the three bits were placed in bit positions 0-3-4.  (Thus producing the numeric values 3, 6 and 9, respectively.)

A variant is the [[United States Postal Service]] [[POSTNET]] [[barcode]], used to represent the [[ZIP Code]] for automated mail sorting and routing equipment. This uses two tall bars as ones and three short bars as zeros.  Here, the weights assigned to the bit positions are 7-4-2-1-0. Again, zero is encoded specially, using the 7+4 combination (binary 11000) that would naturally encode 11. This method was also used in North American telephone [[multi-frequency]] and [[crossbar switch]]ing systems.<ref>{{cite web|url=http://simplethinking.com/photo/phone/vintage/xbr%20relays.shtml|archive-url=https://web.archive.org/web/20060316134506/http://simplethinking.com/photo/phone/vintage/xbr%20relays.shtml|archive-date=2006-03-16|title=Vintage Telephone Equipment Museum — XBR #5 MTF Relay Equip|website=SimpleThinking.com}}</ref>

The USPS [[Postal Alpha Numeric Encoding Technique]] (PLANET) uses the same weights, but with the opposite bar-height convention.

The [[Code 39]] barcode uses weights 1-2-4-7-0 (i.e. [[Least significant bit|LSB]] first, [[Parity bit]] last) for the widths of its bars, but it also encodes two bits of extra information in the spacing between bars. The ||&nbsp;||| spacing is used for digits.

The following table represents [[decimal]] digits from 0 to 9 in various two-out-of-five code systems:

{| class="wikitable" style="text-align: center;"
|-
! Digit
! Telecommunication <br/> 01236
! POSTNET <br/><div align=left>74210</div>
! PLANET
! Code39 <br/> bar widths <br> 12470
|-
! 1
| 11000
| 00011 [[Image:POSTNET 1.svg]]
| '''┃┃┃╻╻'''
| ▮ {{pipe}} {{pipe}} {{pipe}} ▮
|-
! 2
| 10100
| 00101 [[Image:POSTNET 2.svg]]
| '''┃┃╻┃╻'''
| {{pipe}} ▮ {{pipe}} {{pipe}} ▮
|-
! 3
| 10010
| 00110 [[Image:POSTNET 3.svg]]
| '''┃┃╻╻┃'''
| ▮ ▮ {{pipe}} {{pipe}} {{pipe}}
|-
! 4
| 01010
| 01001 [[Image:POSTNET 4.svg]]
| '''┃╻┃┃╻'''
| {{pipe}} {{pipe}} ▮ {{pipe}} ▮
|-
! 5
| 00110
| 01010 [[Image:POSTNET 5.svg]]
| '''┃╻┃╻┃'''
| ▮ {{pipe}} ▮ {{pipe}} {{pipe}}
|-
! 6
| 10001
| 01100 [[Image:POSTNET 6.svg]]
| '''┃╻╻┃┃'''
| {{pipe}} ▮ ▮ {{pipe}} {{pipe}}
|-
! 7
| 01001
| 10001 [[Image:POSTNET 7.svg]]
| '''╻┃┃┃╻'''
| {{pipe}} {{pipe}} {{pipe}} ▮ ▮
|-
! 8
| 00101
| 10010 [[Image:POSTNET 8.svg]]
| '''╻┃┃╻┃'''
| ▮ {{pipe}} {{pipe}} ▮ {{pipe}}
|-
! 9
| 00011
| 10100 [[Image:POSTNET 9.svg]]
| '''╻┃╻┃┃'''
| {{pipe}} ▮ {{pipe}} ▮ {{pipe}}
|-
! 0
| 01100
| 11000 [[Image:POSTNET 0.svg]]
| '''╻╻┃┃┃'''
| {{pipe}} {{pipe}} ▮ ▮ {{pipe}}
|-
|colspan=5|
|-
! A
| 1––10
!rowspan=3 colspan=3| IBM 707x<br/>Sign flags
|-
! −
| 1––01
|-
! +
| 0––11
|-
! Digit
! 01234 <br/> IBM 707x
!colspan=3|
|}

The requirement that exactly two bits be set is strictly stronger than a [[parity check]]; like all [[constant-weight code]]s, a two-out-of-five code can detect not only any single-bit error, but any [[unidirectional error]] -- cases in which all the individual bit errors are of a single type (all 0→1 or all 1→0).

==See also==
* [[Bi-quinary coded decimal]]

==References==
{{reflist|refs=
<ref name="MIL_1991">{{cite book |title=Military Handbook: Encoders - Shaft Angle To Digital |publisher=[[United States Department of Defense]] |id=MIL-HDBK-231A |date=1991-09-30 |url=http://everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf |access-date=2020-07-25 |url-status=live |archive-url=https://web.archive.org/web/20200725051128/http://everyspec.com/MIL-HDBK/MIL-HDBK-0200-0299/download.php?spec=MIL_HDBK_231A.1809.pdf |archive-date=2020-07-25}} (NB. Supersedes MIL-HDBK-231(AS) (1970-07-01).)</ref>
}}

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[[Category:Barcodes]]
[[Category:Computer arithmetic]]
[[Category:Telephony signals]]