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'NJ11TUTE OF STATaS11CI
IOX-5457
ITATE COLLEGE STATION
RALEIGH, NORTH CAROLINA
LOW WEIGHTS
m QUADRATIC-RESlI>UE
CODES*
by
E.F. Assmus Jr,
H.F. Mattson Jr,
R. Turyn
University of North Carolina
Institute of Statistics Memeo Series No. 484.3
August 1966
This research was supported by the Air Force Office
of Scientific Research Contract No. AF-AFOSR-760-65 and
also partially supported by Contract No. AF19(604) -8516.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORm CAROLmA
Chapel Hill, N. C.
* Work done with the cooperation of Prof. J. B. lttlskat of the University
of Pittsburgh w:i:th support of NSF grants GaP•. 2091 and G.P. 2310.
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This is a slightly revised version of the talk presented at the NATO
swmner school in coding in August 1965.
The revisions consist of minor
corrections and the new results for e = 24 recently found by Prof. Muskat.
mTRODUCTION
In order to realize Shannon's theorem on the attainability of channel
capacity, it is necessary to construct codes of longer and longer block length
wi th fixed information rate and good error-correcting properties.
As yet
there is no constructive method for obtaining such a sequence of codes.
In
fact, no one has as yet constructed a sequence of codes of longer and longer
0
block lengths with info~tion rate (kIn) and correction rate (din) bounded
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away from zero.
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residue codes, whose groups of symmetries are known to be large.
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For various reasons there has been hope of finding such a
sequence from among the cyclic codes, in particular from among the quadratic-
The problem of finding the miniinum distance :for the quadratic-residue
codes appears extremely difficult.
only fragmentary nontrivial results.
So far there have been (to our knowledge)
The (23, 12) code of Golay over GF(2),
is, of course, well-known to have mininn.un distance d· = 7.
that d = 11 for the (47,24) code over GF(2).
Gleason has shown
Mattson and Prange have shown
the (41,21) code over GF(2) to have d = 9 and Pless and Prange have shown
~he (71,36) code over GF(2) to have d = 11.
We have recently shown that the (19,lO) code averGF(4) has d
tl-at the (23,12) code over GF(3)hasd
=7
and
= 8. *
OUr purpose here is to describe °a method of finding vectors of low
0
we1ghtsin quadratic-residue codes.
nature discouraging.
Thus, the following results are by
The method, however, will
~
find low-weight vectors
*. The (3l{16) code over GF(2) has, d=7; the (17,9) code over GF(2) has d=5;
the (11,6) code and (13,7). code, both over GF(4), have d=5.
only nontrivial determinations of d we are aware of.
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These are the
for those quadratic-residue codes of block leDgt;h congruent to
-1 modulo
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Instead of s1v1n1 P1'OOfs for our results, we have tried to defiDe IU1d
describe tbeJll in such a W&:1 that people not :tam:1liar with the number theo1'7
.
used D&7 nevertheless bave some UDderstaDd1ug of what we have done.
A quadratic-residue code is defined u
length A over the field QF(q) w1th generator
a cyclic code of prime block
po~m:1al of
the form
g(x) = n (x - ~r) or (x-l)g(x), where R is the set of quadratic residues
r~·
.
th
modulo A, aDd ~ isa pr1m:Ltive A root of lover GI'(q). !he dimension of
such a code is (A
.:t 1)/2,
and there are two such codes of each of these di-
mensions provided only that q is a quadratic-residue modulo A. The two codes
6f each d1JMDs1on are equivalent to each other under the permutation i ... ti,
i = 0, 1, ••• , A-l, of coordinates Jlt)d A, were t is any fixed elemeDt of R',
the set of quadratic DOnres1dues.
.
We denote the (A, (A-l)/2) codes b7 A aDd
B and the (A, (A+l)/2) codes by A+ and B+, where we take A+ ::> A and B+ ::> B.
2
.
It is necessary to consider only the' cases q = P, aDd q = p 1dJen p is
•
in R', since quadratic-residue codes over ar(pn) are obtained from those
over ar(p), or Gl(p2 ), mere~ by taking tensor products (extending the co-
efficient field); aDd we know this process does not change the m1.n1npJID
weight.
If d and
d:L
are the minimum distances in A+ and A, respective17, then
results of Gleason &ad Prange show tbat ~ • 1 + d. 1'hus in particular the
minimum weight in A+ is not achieved by a vector in A.
The square-root bound on d is derived as follow.
+
.
A po~m1al ~ (x)
+
of weight d in A IIII1ltipli.ed by one ~(x) of the same weight in B JIIl18t give
a constant JII11tiple of x A- l + ••• + 1. When A & -1 (JIt)d l,.), we _y take
~(x) = ~ (x-l ), and then it is immediate that d(d-1) ~ A-1. When J B +1
2
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(ood 4), we conclude ;
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To our knowledge the square-root bound is the best lower bound on d for
quadratic-residue codes.
There is some hope that it may not be a very good
lower bound, and that among these codes one could find an infinite sequence
of values of
I,
such that for some fixed p the associated codes A had d/ I,
bounded awa:y from O.
This paper reports on an attempt to find out more about the minimum
weight in quadratic-residue codes.
The results are somewhat discouraging,
because in some cases d/.1J was found to be lower than expected.
Previously
d/" was known to be as small as about 1/10 in some cases; this study finds
that it can be as low as about 1/24.
BAClGROUND
We now introduce some elements of algebraic nwnber theory in order to
explain our method.
denote a primitive
Let Q denote the field of rational nwnbers and let z
I,th
root of 1, where
One may think of z as exp(2ri/I,); z"
=1
is an odd prime number as before.
I,
and no lower power of z is 1.
Let
lC denote the field Q(z); K consists of all "numbers" of the form ao + alz +
... + a,,_2zl,-2 where
80 ,
0
... ,
801,_2 are in Q.
taining Q and z, and zl,-l + z.e-2 + ••• + 1
lC is the smallest field con-
= O.
Therefore any 1,-1 distinct
powers of z form a basis of K as a vector space over Q.*
Any "-1 distinct po'trers of z form an integral basis of lC/Q.
For the rest of this paper, let p be a rational prime, p
p is in R.
Then the (J"
r
1"
such that
(.1J+l)/2) quadratic-residue codes over GF(p) will be
* At
this point the reader unfamiliar with number theory may wish to read 'khe
- Appendix.
3
B;.
denoted by ~ and
The order h of p under multiplication DX>dulo .I must
The polynomial x.l-l + x.l-2 +
divide (.t-l)/2, so that g in hg = .1-1 is even.
• ~. + 1 splits over GF(p) into the product' of g distinct irreducible factors
d (x),
'1'0
••• , ¢ g_lex) each of degree h.
The prime p splits in Ie into the product of g distinct prime ideaJ.s
p,
o ••• , P g-l' which can be labelled in such a way that (Z denoting the
rational integers)
LEMMA. For
.1-1
any j, an integer a =
I
of Ie, a
i
E:
Z, is divisible
o
.1-1
by Pj if and only if the polynomial
\'
L
t
i
aix is divisible by ¢j(x) over
o
f
GF(p), 'Where a
i
denotes the residue-class of a
i
modulo p.
The Galois group G of K/Q is cyclic of degree t-l.
If
(f
denotes a gen-
erator of G, it is convenient to label the Pi and ¢i (x) for any pas follows:
Choose Po arbitrarily and let ¢0 (x) be its corresponding polynomial.
define Pi
t
= ;(po)
and call ¢i (x) the polynomial with
is a root of ¢o (x).
Then
~(f-i as a root; where
Then Pi and ¢i (x) correspond under the I.enmm..
Under
this labelling and correspondence, the generator polynomials of the quadratic+
+
f
residue codes A· and Bp over GF (p) correspond to n = p P2" P 2and:rr =p p ••• p
,
p
0
g.
1 :>
g-l
f
The ideUs :rr and
1£
are the prime ideals into which (p) splits in the
quadratic field L contained in K (and this notation will be used throughout).
L is the fixed field of ;..
I zt(t~t).
T} f
=
.f-
+ x + (1
.:!: .1)/4,
It is generated by
The irreducible polynomial of
I
=
T}
T}
and
zr (r~) or by
T} t
over
Q
is
the sign being· chosen so that the constant term is in Z.
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Thus L is Q(J"z.t), where this sign is the same as that in the congruence
~ ==
.:t
~-l
I a~
1 (IOCld 4).
What we have observed is that when (p) = 1ur' in L, the ~ynOm:i.al
xi over GF (p) is in
~
or B; if and only if the integer
o
La i zi of X is
0
divisible by
1f
or
1f',
,
where a. is the residue-class of a. IOCldulo p.
~
~
It is not easy in general to tell whether an integer of X is divisible
by
or
1f
1f',
but in the ,special case where the integer is a so-called
cyclotomic period it is a feasible computation.
We shall use the following criterion.
PROPOSITION 1.
Let
e be
an integer of the field E lying between L and
and let T denote the trace from E to L.
Ie,
gcd (Te , dl'e ).
Then e is divisible by
1f
Suppose that p is prime to the
or
1f'
if and only if e· ;( e) is
divisible by P for each odd i.
The usefulness of this criterion is that in some cases, including that
in 'Which e is a cyclotomic period, it is easy to express e cri (e) in the form
.e-l
!.
i
a. z ; and this is divisible by p if and only if p divides each a .•
~
~
1
t-l
(When
e=
I
i
a i z , the condition is that a
o
== a
l
== ••• ==
o
CYCLOTOMIC PERIODS
Let t-l
f.
= ef.
Then the Galois group G has a unique SUbgroup H of order
By means of the action of G on the powers of z we identify G with the
cyclic group of non-o residues IOCldulo t under multiplication.
the SUbgroup of eth powers of G.
Hi IOCldulo H as G = H U
o
HJ. U
H is then
We nay decompose G into a union of cosets
••• U H _ and we then define the cyclotomic
e l
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periods for e as
TJi =
L
zS
i ;:: 0, 1, ••• , e ..l,
s€H
i
'Where we choose the labelling so that"
I
=
,
zS(se:H) and TJ
i
= <tiC,,).
The cyclotomic periods for e are an integral basis for the field E over
Q, where E = Q( TJo ) is the fixed field of H.
Therefore for each i = 0,1, ••• ,
e-l there exist rational integers denoted by (i,O), (i,l), ••• , (i,e-l) such
that
e-l
TJoTJi =
I
(1)
(i,j)TJ j + Eif
j=o
where
6. ;::
J.
wise Ei
1 'When f is even and i = 0, or when f is odd and i
= O.
= 1 'if
(In other words, Ei
= e/2;
other-
and only if -1 is in Hi.)
It follows that
e-l
L
(i,j) + E.
J.
=f
(2)
for all 1.
j=o
These non-negative rational integers (i, j) are called the CIclotomic numbers
o! order e.
If w is the primitive element n>dulo J, corresponding to
(i,j)is the number of solutions (x,y) with 0
wex+i + 1
= wey+
j
=s x,
(mod
(j,
y < f of
t).
It is from this relation that Prof. t41skat has calculated the (i, j) for
various e and t.
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then
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The cyclotomic numbers satisfy
= (-i,j-i)
(i,j)
and (i,j)
={
(j, i)
f even
(j+e/2, i+e/2) f odd,
where the entries are read mdulo e.
AN APPLICATION TO QUADRATIC-RESIDUE CODES
From what we have seen, every time that \, is divisible by
there is a vector of weight f = (1-1)/e in A+.
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jft,
Let us call T}o (x) the polyT}o.
To see that the criterion
of Proposition 1 holds, we note that TT}0
_
= T}
xS(seH).
We sha.1l now apply the foregoing to the integer
,
(j
TT}o =
T) ;
I
T} + T}
= -1.
Therefore \, is divisible by
only if T} T}. is divisible by p for each odd ,i.
o
(j
or
e=
nomial
and
1C
J.
1C
or
~I
if and
The choice of the generator
determines the labelling of T}l' ••• , T}e-l' and the cyclotomic numbers
change as
(j
changes; but "Whether
1C
or
jf'
divides \, is independent of the
choice of generator, hence so is our criterion expressed in terms of cyclotom1c numbers.
In terms of the cyclotomic numbers this condition is that
for each odd i we must have, from (1),
(i,O) ==
p).
(i,l) = ••• == (i,e-l) == e.f(IlX>d
J.
(3)
Some necessary conditions are:
PROPOSITION 2.
p divides (.t
if e
If \, (x) is in
A; or B; for some p, then e is even and
:!:. 1)/4, the sign chosen to make this a rational integer. And
> 2, then p divides f, which implies that T}o (x) is in Ap or Bp •
7
Thus in searching for code-vectors by this means we restrict ourselves
And furthermre, we have
to the ca.se when e is even.
COROLLARt.
Let /. == 1, (mod 4), and let t-l
cyclotomic period
= ef
with e > 2.
110 of order e never satisfies 110 (x) ~ or B; for
In view of this we restrict ourselves to the case
even, e
> 2.
Then the
In this case
E.
J.
any p.
t == 1 (md 4) and e
= 0 for all odd i, so that (3) becanes
°
Let /.-1 = ef == (mod 4), e even, e > 2.
Then p divides (i, j) for all j and all odd i *
if and only if 110 (x) is in \ or
a cyclotomic period of order e.
.B;, where Tlo
is
The following resu!t is a helpful sieve when one is searching for such
code-vectors.
PROFOSITION 3.
Let t-l = 2ef with e even.
Then the cyclotomic period
Tlo for 2e cannot yield a code-vector Tl (x) in A; or B; for a given p unless
o
a cyclotomic period of order e does so (far the same p).
There exist formulas for the (i,j) for certain values of e.
=4
for e
t
and f even, Gauss proved that 16(1,0)
2 + 4y2 with x == 1 (md 4).
= x
formulas of
lOOre
= t-l
For exa.II!Ple,
+ 2(x-l) + By, where
There are similar but IOOre complicated
recent date for some other e up to e
= 20.
But for our
pur:poses, 'Which are to find examples of quadratic-residue codes having
vectors of rather low weight, it is more appropriate to search through ta.bles
of values of the (i,j) for instances of (4) than to establish general conditions with the formulas and then compute solutions of t = (some quadratic
form).
* This
condition -implies that p is a quadratic residue modulo/. , since p If.
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For purposes not related to coding, Prof. Mlskat has calculated the
cyclotomic numbers for all primes less than 50,000 for various values of e
up to e = 20.
He very kindly gave us tables of these values, from which
we found several instances (10 < e < 20) where TJ (x) was in A+ or B+.
-
-
Some
p:p
0
examples of these findings are:
e=10,
feven
f even
!
e=14,
e=16
e=20,
e=24,
f odd
f even
f even
f odd
f even
!
!
1.
e=12
E.
1. R.
761 2 2,521 2 8,221
1,361 2 8,089 2 18,541
1,381 3 47,521 3 22,621
6,841 2,3 49,681 2 25,261
8,821 3
44,269
21,401 5
30,941 7
49,921 2
N =
94
N = 81
N
E.
E.
E.
!
E.
5 1,289 2 4,801 2 P, = 8821,
9,601 2
3
6,301 3 15,937 3 P = 3, is the 18,433 2
5 49,057 2 47,713 2 only instance. 48,817 2
5
= II
7
N = 27
N = 13
N=l
N
=7
N is the number of primes less than 50,000 in the table in question giving
an instance where TJ (x) is in Ap or Bp • The values of N for e = 10, 12, 14,
o
and 24 were found on his computer by Prof. Mlskat. We obta.ined the values
for e
= 16
and e
= 20 by our own
visual search of his tables (the same for
e = 14, where our result agreed with his).
The smallest and largest values
of P, for each value of e are given, as well as an
e~le
for each value
of p that arose.
When f is even, a large majority of the instances occur for p = 2;
when f is odd, occurrences are rarer.
We also calculated several cases by hand for smaller values of e, using
general formulas; in the same way we verified the entries in Prof. ltfu.skat f s
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tables for three values of J" namely,
p,
= 1321 and 1657.
9
J,
= 3,821, e = 10; and for e = 12,
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SUMMARY
To repeat:
each instance we found yields a vector of weight (.t-l)/e
in Ap or Bp e In each case these weights are far greater than ~.t, but there
is no reason to suppose that the weights (.t-l)/e are the
minjnnlm
weights in
the J\,e
* (approximate),
The lowest weight-to-block length ratio found, namely 1/20
is lower, by a factor of about 1/2, than the lowest such ratio we knew of
before this investigatione
The fact that this approach cannot yield low weights in ~ when .t=4n-l,
plus similar resutls below, holds out a faint gliImner of hope for a sequence
of ~ with non-vanishing error-correcting rate in this class of codes.
Perhaps a word on the role of conqmtation in this method is called for.
One could get our results directly by first factoring x t - l + e' e + lover
~
.
GF(p) into its two "quadratic-residue" factors ')'1(x) and ')'2(X)' . and then
s
computing whether ')'i (x) divides
x , ~ere the sum is over s€H, and 'Where
L:
it would usual1.y be necess·ary to compute this for both i
=1
and i
= 2.
By
the previous analysis one could restrict oneself' to those p dividing f.
Although we have not made careful estmtes, it appears that perhaps the
amount of "direct" computation required for a given J, and e (ef
determine 'Whether
L:
= t-l)
to
x s (s €H) is in. A; or B; for all p dividing f might be
about the same as that required to calculate the cyclotomic numbers of order
e for .t.
Perhaps the latter calculation would be significantly easier.
In
any case, the fact that the cyclotomic numbers have been studied since the
time of Gauss, and the fact that those used by us were calculated by Prof.
Mlskat for a purpose entirely different, lead us to regard this approach as
{-i"lNow 1/24, as of early 1966.
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less computational than the direct approach •
CODE-VECTORS FROM OTHER mTOOERS
We may consider by this approach any integer
I.
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C+
of the field E lying between K and L, of even degree e over Q.
A sufficient condition for the criterion of Proposition 1 to obtain is
I
that p should not divide the gcd(ce -
ck '
I
C2i -
I
c 2i +l )·
We define e(x) in analogy with the previous definitions, and if Proposition
1 holds for e, then e(x) is in ~ or
C(C j + Cj _i ) +
I
B; if and only if for each odd i we have
ckcm(i+k-m, j.-m) -
c
2
k,m
+ f
L.
€i+k_mCkCm(mod p),
k,m
j = 0, 1, ••• , e-I.
Ie
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I
I
e=
A necessary condition for (5) to hold is
This is obtained on adding (5) over j.
In particular this implies that
Lk,m €i+k_mckcm is constant modulo p for odd i; therefore if only one ck
is non-o, and if
J,
== -1 (md4), this condition cannot hold for even e
> 2.
Another necessary condition is obtained on multiplying Te and cf.['e:
Let E denote
L
c 2i and I denote
L.
c 2i +l ; then e(x) is in \
11
or
B; implies
I
I
(ce/2)
where the
2
+ IE(l -
:t signs
:J.;!.:,t
T )-
(ce/2)
\
L ck
+ (E
2
-2
+ r-)
l.:!:.t
T
== 0 (mod p)
are the same and are chosen to make (1;tJ)/4 in Z.
Obviously the condition (5) is nx>re complicated when 9 is a sum of IOOre
than one cyclotomic period.
We have not searched very hard for code-vectors
of this type, but we did find some instances where 9(x) was in
e = 14, f even, and e = 16, f even, for 9 = 1 + T'l.
o
9
=1
+ T'lo for e
= 20,
A; or B; for
There were none for
f odd, however.
One is less interested in these nx>re complicated 9's because they would
give vectors of higher weights in the codes; on the other hand, it might be
possible to use these 9' s to find vectors of weight very near the minimum.
weight if one tried several sums for large values of e.
For example, when
J = 29 we could take e = 14 and look for code1."V'ectors 9(x) arising from
9 = c + CoT'lo + ... + C T'l3 over p = 5 (or we could take Q as a sum of some
3
other four periods). This would give weight 8 or 9 depending on whether c = 0
or not.
(The minimum we~ght is at least 6 and is probably 8 or 9.)
As a further example, T'lo(x) (of weight 6) is in ~ for
.t = 13; this code
has minimum weight at least 5 and probably 6.
We plan to pursue this question in the future.
APPENDIX
In the rational integers Z every non-Q element is uniquely expressible
as a product of prime numbers, apart from the ordering of the factors.
This
property holds IOOre generally in fields of algebraic numbers such as the
field Ie = Q(z) considered here.
One can define a subring IS). of le called the
integers of le, but in general one needs to introduce ideals in order to get
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unique factorization.
The integral ideals are sUbrings of
closed under multiplication by elements from
6J.
~
which are
One multiplies ideals !.
and !? by taking the set of all sums of the products ab, with a €!. and
The resulting set !. •
b €!?.
E.
The sum!. +
is also an ideal.
as the set of all a + b with a €!. and
b €
E..
E.
is defined
Under these definitions the
ideals form a ring containing an identity eleJJM:!nt t9.
Certain of these
ideals are called prime ideals, and they have the property that every ideal
can be expressed uniquely as a product of prime ideals.
C9
For the fields lC:::> L:::> Q with rings of integers
:::> tY
L
:J
Z, we know
that the ideal in L of the rational prime p, namely the set p·&U is either
itself a prime ideal of 6'L (when p is a quadratic nonresidue modulo .8) or is
the product of two different prime ideals
quadratic-residue mod .e).
this case the ideals
1f •
1f
and
1f'
of &L (when p is a
It is the latter case that we assume here.
t9 and
1f'
•
In.
t9 consist of the product poP2 ••• Pg-2
and P1P3 ••• p g-1 of the prime ideals p.J. defined in the text, and each p.J.
corresponds to a different irreducible factor
~.J. (x) of x.e-l + ••• + lover
GF(p) •
We say that an integer of lC is divisible by a given ideal if it is an
element of that ideal.
The point of our method is that, for an integer
.e-l
.e-l
Q
=
I
a zi of lC, a i e:Z, the associated polynomial Q(x) =
i
is divisible by ~o(x) ~2(x) ••• ¢g_2(x) =
rr(x_~t), t€R
o
g-2 =
class of a mod p.)
i
by p P2 ••• P
aixi over GF(p)
o
o
... ¢g-l (x) =
I
1f (9
f
,
rr(x_~r), r€R, or by ¢l(x) ¢3(x)
if and only if the.. integer 9 of lC is divisible
or by P1P3 ••• P
13
g-1
=
1f'
&.
(a! is the residueJ.
Thus we are able to use information about integers of K contained in the
cyciotomic numbers to give us some information on the quadratic-residue
codes.
For :further explanation of the points sketched in this appendix, the
reader is referred to any book on algebraic number theory, such as, for
example, E. Heeke, Algebraische Zahlen, (Chelsea, New York, 1948),H. Weyl,
Algebraic Theory of Numbers (~inceton, 1940), or H. B. Mann, Introduction
to Algebraic Number Theory: (Ohio State, 1955).
A reference on cyclotomic numbers is L. E. Dickson's "Cyclotonw,
Higher Congruences, and Waring's Problem, II Am. J. Math.; vol. 57, pp. 391-424;
1935.
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