The purpose of a tail sequence for command link transmission units is to fail to decode, so that the command decoder will begin searching for the start of the next unit. A tail sequence used by several missions and recommended for this purpose by the Consultative Committee on Space Data Standards is analyzed. A single channel error can cause the sequence to decode. An alternative sequence requiring at least two channel errors before it can possibly decode is presented. (No sequence requiring more than two channel errors before it can possibly decode exists for this code.

Dolinar, S.

Ekroot, Laura

Mceliece, R.

Swanson, L.

English

NASA Technical Reports Server

S93-29596
TDA ProgressReport42-113 May 15,1993
r/
Uncorrectable Sequences and Telecommand
L. Ekroot, R. McEliece, 1 S. Dolinar, and L. Swanson
CommunicationsSystemsResearchSection
The purpose of a tail sequence for command link transmission units is to fail to
decode, so that the command decoder will begin searching for the start of the next
unit. A tail sequence used by several missions and recommended for this purpose
by the Consultative Committee on Space Data Standards is analyzed. A single
channel error can cause the sequence to decode. An alternative sequence requiring
at least two channel errors before it can possibly decode is presented. (No sequence
requiring more than two channel errors before it can possibly decode exists for this
code.)
I. Introduction
When a command link transmission unit (CLTU) con-
sisting of many codeblocks is received by a spacecraft, the
command decoder verifies that each codeblock is a valid
codeword and accepts it, or that it is a slightly corrupted
codeword and corrects it, or that it is too far from a valid
codeword and rejects it. Rejecting a codeblock causes the
receiver to give up on the unit and begin searching for the
start of the next unit. At the end of the CLTU, there
is a tail sequence designed to be rejected as a codeword,
sending the decoder into a "search mode." This article an-
alyzes the performance of the tail sequence recommended
by the Consultative Committee on Space Data Standards
(CCSDS) and used by several missions. So instead of the
1Consultant, California Institute of Technology, Engineering De-
partment.
usual question about a code, i.e., how many errors can the
code correct or detect, the question here is how many er-
rors can occur before an uncorrectable sequence becomes
correctable.
II. Analysis of Uncorrectable Sequences
In order for a sequence to be uncorrectable, it must be
far enough from a codeword to cause the decoder to not
decode. At the very least, it must differ from the near-
est codeword in more positions than the decoder is able
to correct. However, channel errors can make such a se-
quence decodable. The more errors that must occur before
the sequence becomes correctable, the less likely it is that
the sequence will accidentally decode. In order to maxi-
mize the number of channel errors before the sequence will
decode, it is necessary to characterize and find sequences
that are as far away from codewords as possible.
99
https://ntrs.nasa.gov/search.jsp?R=19930020407 2020-03-17T04:29:56+00:00Z
A code is designed to have codewords that are maxi-
mally far away from each other. If, as a simple example,
a new code is created by using a subset of the codewords,
then the unused codewords are still far away from the code-
words of the new code. Intuitively, these unused codewords
are candidates for uncorrectable sequences.
The codes discussed in this article include the perfect
(63,57) Hamming code, the (63,56) expurgated Hamming
code, and shortened versions of the expurgated code. The
codewords for the shortened codes correspond to subsets
of the codewords of the (63,56) code, which are themselves
the even codewords from the (63,57) code. By making use
of the larger code's properties and keeping track of what
happens as the codeword sets get smaller, sequences that
are uncorrectable in each of the smaller codes can be found.
A. The Perfect (63,57) Hamming Code
The generator polynomial for the (63,57) code is the
sixth-degree primitive polynomial
gp(x) = x6 + x + 1 (1)
The (63,57) code is perfect and has minimum distance 3,
i.e., every binary sequence of length 63 is either a code-
word or is Hamming distance one away from exactly one
codeword. For a perfect code, there are no holes left when
the space of binary sequences is filled with balls of radius
one centered at the codewords.
The even-weight words are a promising subset to use for
a new code, leaving the odd-weight words as candidates for
uncorrectable sequences. In fact the (63,56) code described
in the next section uses the even-weight words.
B. The (63,56) Expurgated Hamming Code
The generator polynomial for the (63,56) code is given
by
g(x) = x7 + x6 + x 2 + 1 = (x + 1)(x 6 + x + 1) (2)
Since the generator polynomial is the product of z + 1
and the generator polynomial gv(x) of the perfect (63,57)
code, the (63,56) code consists of only the even-weight
codewords from the perfect (63,57) code. This code has
minimum distance four and can correct at most one error.
Note that any odd-weight codeword in the (63,57) code
is exactly Hamming distance three away from the near-
est even-weight codewords, and the even-weight codewords
are the codewords of the (63,56) code. Any odd-weight bi-
nary sequence of length 63 differs from a nearest codeword
by either three bits or one bit. Similarly, any even-weight
sequence is either a codeword or is Hamming distance two
from the nearest codewords.
The following example illustrates how the concepts of
distance relate to a sequence that is not a codeword in
the perfect (63,57) code. The sequence selected for the
example plays a role in the CCSDS recommendations for
telecommand; the description of that role is deferred to
Section III.
Example 1: The length 63 sequence
01010101010101010101010101010101010101010101010101010101 1010101
has even weight, and is therefore either a codeword in the (63,56) code or two away from a codeword. The two-part
syndrome _,s2(r)] of a sequence r = rN-lrg-2""rlr0 is given by
( _-_,-o r,a (m°d a + a + l ) )
, 6
- N-_ i (3)k.s2(r)] = _,i=o rla (mod a + 1)
where a is a root of the generator polynomial g(x), and N is the length of the codewords. The two-part syndrome
tells if the sequence is a codeword, how to correct it if it is distance one from a codeword, or that it is not near enough
to a single codeword. Specifically, if the syndrome is (0°), then the sequence is a codeword; if the syndrome is (0), then the
10o
sequence has odd weight and differs from the nearest codeword in three positions; if the syndrome is , the sequence
has odd weight and differs from the nearest codeword in the jth position; if the syndrome is (o _), the weight is even
and the sequence differs from the nearest codewords in two positions.
For this example sequence, the top part of the syndrome (modulo a e + a + 1) is
N-1
sa(r)_ _r,a i (moda 6+a+l) (4)
_=0
(5)
1 + a2 +O_4 + (a + 1)+ --
a7 + _s3
1 +_2
(6)
= a + a 2 + a 4 + --an6 + 1 (7)
l+a 2
_(_ + 1)+ 1
= (_+ _2+ _')(I+ _2)+ (s)
- 1 + a 2 1 + a 2
a._-a2-[-o_4-_-a3-_-o_4+a6-_-a2-[-o_+ I
= (9)
l+a 2
c_3+ _6 + I
= (10)
l+a 2
= a3+(l+a)+l (11)
- 1+(_
cz3 -.].-_
= _ (12)
= a (13)
where Eqs. (6), (8) and (11) follow from equivalence modulo a 6 + a + 1. The bottom part of the syndrome (modulo
_+1) is
N-1
s2(r) = Z rlai (mod _+ i) (14)
i=0
= weight(r) (mod 2) (15)
-32 (mod 2) (16)
- 0 (17)
lOI
The syndrome (o) indicates that the sequence is not a codeword in the (63,56) code, and is two away from the nearest
codewords. A single error in any bit except rl will make the sequence differ from a codeword by one, and thus decodable. 1
If instead a sequence is considered that is an odd-weight word in the perfect (63,57) code, it is three away from the
codewords of the (63,56) code. This means that two errors must occur before it becomes decodable by a single-error-
correcting (63,56) decoder. Such a sequence would be a better choice for a tail sequence because it is more resistant to
accidental decoding in the presence of errors.
In selecting a particular sequence for the command coding application, the effects on the distance and uncorrectability
properties as the code is shortened must be taken into account.
C. Shortening the (63,56) Code
Select a subset of a code, where all the codewords in the subset have zeros in some specified positions. Since all of the
codewords in the subset have zeros in the specified positions, those positions carry no information and can be ignored.
The resulting set of codewords forms a shortened code. Shortening cannot decrease the minimum distance, and will only
increase the minimum distance if the code is shortened severely. _
For this application, shortening will be done by taking only the codewords which have zeros in the leftmost or first
rn positions. Note that for each shortened word there is a corresponding full-length word that has zeros in the first m
positions.
Example 2: Consider the sequence of length 55
010101010101010101010101010101010101010101010101 1010101
It corresponds to the full-length sequence
00000000010101010101010101010101010101010101010101010101 1010101
with weight 28.
The top part of the syndrome for the full-length sequence is
N-1
sl(r)-- Zr'(_' (moda 6+a+l) (18)
i=0
= 1 -{- 0_2 + 014 -l- _6 -I- 0_7 -I- or9 -}- 0_11 -I-' "'-{- a 53 (19)
-- 1 + a s + a4 + (1 + a) + (o__ + a 9 + Ot 11 "Jr ''' "[- _53) (20)
_=a+ a2 + a4 + a(a + 1)+a(a + 1)9 (22)(1+
1 If the error is in rl, the syndrome becomes (0), indicating that it is three away from a codeword.
2 For the (63,56) code, it can be shown that as long as the shortened length is greater than 32, the minimum distance will remain 4, and the
maximum distance of any sequence to the nearest codewords will remain 3.
102
= c_+ a2 + c_4 + a(1 + (c_+ 1)8) (23)
l+a
_(1 + c_s + 1) (24)
=c_+a2+a4+ i+o_
_ ¢_+ _2 + _4 + _a2(_ + 1) (25)
1+¢_
= Ol -_ 0¢ 2 -_- O_4 -'[- 0¢ 3 (26)
= c_((_ + 1) 3 (27)
O_ 19 (28)
(°:)The bottom part of the syndrome is zero since the weight is even. The two-part syndrome indicates that the
sequence is not a codeword, and that it is two away from the nearest codewords. A single error can make the sequence
differ from a codeword by one, and thus deeodable.
D. Finding a Good Uncorrectable Sequence
The concepts in Section II.B and Section II.C lead to the definition of a good uncorrectable sequence as one for which
it and all the desired truncations of it are maximally distant from the codewords in the corresponding code, i.e., an
odd-weight codeword in the perfect (63,57) code. The syndrome for good uncorrectable sequences is (lO). The analysis
below shows that an uneorrectable sequence can be chosen so that, when it is truncated by octets, its syndrome does
not change, and therefore it does not become correctable.
If a given sequence is truncated by m bits, and if it is desired that the syndrome not be changed by the truncation,
then the smallest possible m is eight. This is because in order to not change either part of the syndrome, the truncated
bits must correspond to a polynomial that is zero modulo both o_+ 1 and er6 + c_+ 1 ; the lowest order nonzero polynomial
satisfying that requirement is g(c 0 = c_7+ c_6+ c_2+ 1. Therefore the shortest such nonzero sequence is 11000101, which
has length eight.
There are engineering reasons for truncating by octets in the application considered in Section III. Also, bit syn-
chronization requirements often make it preferable to have many transitions in the sequence, and thus a mostly zeros
sequence is undesirable. For the remainder of the article, it is assumed that shortening will be done only by multiples of
eight bits, and that octets of zeros are not of interest.
Since the bottom part of the syndrome must be 1 for the sequence and all of its truncations, and the truncated bits
11000101 have even weight, the nontruncated part of the sequence must have odd weight. Since the top part of the
syndrome must be 0 for the sequence and all of its truncations, the nontruncated part of the sequence must correspond
to a polynomial which is zero modulo a 6 + a + 1. The shortest such sequence is 1000011.
A simple construction of a good uncorrectable sequence is a concatenation of octets of the form 11000101 with the
seven bits 1000011 in the rightmost positions. This is not the only good uncorrectable sequence, but it does have good
distance and bit synchronization properties. The syndrome for this sequence is confirmed in the next example.
Example 3: Consider
11000101 11000101 ll000101 11000101 11000101 11000101 ll000101 1000011
103
This sequence has weight 31, so the bottom part of the syndrome s2(r) is 1. The top part of the syndrome (modulo
a s + a + 1) for the full-length sequence is
N-1
sl(r)-- _r,a' (moda 6+a+l) (29)
i=0
6
= l+a+a 6+_a7+sk(l+a 2+a 6+a7) (30)
k=O
6
- 1 + _ + _6 -I- _-_ av+s_(1 + _ + a6)(1 + a) (31)
k--0
= 0 (32)
Thus, the sequence is three away from the nearest codewords. A single error cannot make this sequence decodable.
III. Command Link Coding
The CCSDS recommendation uses a tail sequence that is specially constructed to not decode. By not decoding, it
causes the receiver to begin searching for the next CLTU. It will be shown that the sequence that has been recommended
is not the best sequence in terms of distance, and a better one will be given.
In order to apply the results of Section II to the command coding problem, the operations of the CLTU must be
detailed. Refer to [1] for more detailed explanations of what is summarized here.
A. Telecommand Codeblock
The telecommand codeblock has K information bits, 7 inverted parity check bits, and a fill bit for a total length of
L. Because of the code selected and the desire to shorten in units of 8 bits, the number of information bits and block
lengths considered are K = 32, 40, 48, and 56, and L = 40, 48, 56, and 64, respectively.
B. Command Link Transmission Unit
The CLTU consists of
(1) a 16-bit start sequence, namely 1110101110010000,
(2) a number of telecommand codeblocks (the information may be padded with fill to make the information into a
multiple of K bits), and
(3) a tail sequence which is a sequence of bits the same length as a codeblock and designed to be uncorrectable. The
idea is to cause the receiver to stop decoding and begin looking for the start sequence of the next CLTU
C. Tall Sequences
The CCSDS recommendations specify that the tail sequence t be a sequence of alternating zeros and ones beginning
with a zero. Ignoring the fill bit, and noting that the parity bits are inverted, it can be seen that for a block length L of
64, this corresponds to the sequence in Example 1. As illustrated in the example, this sequence has distance properties
such that a single channel error can, and almost certainly will, make the sequence decodable.
104
For the shorter block lengths L = 40, 48, and 56, the sequences of alternating zeros and ones beginning with a zero
have corresponding full-length sequences with zeros filled in the first positions. All three of these corresponding sequences
have even weights, and have nonzero syndromes. The calculation for L = 56 is done in Example 2.
If instead the sequence
11000101 11000101 ll000101 ll000101 11000101 ll000101 11000101 0111100 0
is used as the tail sequence, it corresponds (when the fill bit is removed and the parity bits are inverted) to the sequence
in Example 3. If one channel error occurs, then this sequence will still be uncorrectable.
D. Augmentation Using the Fill Bit
The CCSDS recommendations propose to use the fill bit as a flag to tell the decoder to operate in error-detect mode
only. This augmentation is only suggested for use with the tail sequence. This improves the probability of spotting
the tail sequence by not allowing the decoder to correct any errors when the fill bit is 1. In this mode, two errors are
sufficient to make the CCSDS tail sequence (of alternating zeros and ones) decodable, while a minimum of three errors
is required to make the sequence presented here decodable.
IV. Conclusions
The analysis in this article shows that there are uncorrectable sequences that can tolerate one more channel error
than the CCSDS tail sequence before becoming decodable. It is also shown that this property may be preserved for the
shortened as well as the full-length codes recommended by the CCSDS. A sequence satisfying these requirements should
be considered for the role of the CCSDS tall sequence since it is more resistant to channel errors than the proposed tail
sequence, and still has many transitions to aid bit synchronization.
Tables of probabilities of missing the tail sequence, the operation of the decoder on the shortened codes, and the
proof that no sequence requiring at least three channel errors before it can possibly decode exists for this code may be
the subject of future work.
Reference
[1] NASA, Consultative Committee on Space Data Standards, Telecommand Part 1
Channel Service, Blue Book, 201.0-B-I, Washington, D.C., January 1987.
105


Uncorrectable sequences and telecommand

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server