Delay Invariant Convolutional Network Codes
Qifu Tyler Sun, Sidharth Jaggi, and Shuo-Yen Robert Li
Institute of Network Coding
The Chinese University of Hong Kong
Hong Kong SAR, China
qfsun@inc.cuhk.edu.hk, {jaggi, bobli}@ie.cuhk.edu.hk
Abstract—In this work, we define delay invariant convolutional
network codes which guarantee multicast communication at
asymptotically optimal rates in networks with arbitrary delay
patterns. We show the existence of such a code over every symbol
field. Moreover, the code can be constructed with high
probability when coding coefficients are independently and
uniformly chosen from a sufficiently large set of coding
operations. On the other hand, if the symbol field is no smaller
than the number of receivers, we devise a method to efficiently
construct a delay invariant convolutional network code with
scalar coding coefficients.
I.
INTRODUCTION
Convolutional network coding (See [8], [10], and [2]) is a
form of linear network coding which deals with a pipeline of
messages as a whole rather than individually. It allows coding
over the combined space-time domain. Over a network with
cycles, the propagation and encoding of sequential data
symbols may convolve together. Thus the propagation delay
becomes an essential factor in network coding and hence
convolutional network coding is naturally adopted to ensure
causal data propagation around cycles. Under the formulation
of propagation delay in [10], the transmission delay over
channels is absorbed into processing delay at nodes. The node
processing delay, on the other hand, is measured on every
adjacent pair (d, e) of channels, where d and e are, respectively,
the incoming and outgoing edges of a node. The delay over (d,
e), which is a nonnegative integer power of D, represents the
latency from the time a data symbol is received from d till the
time it is first incorporated into the information sent onto e.
Consequently, when data is propagated over the network via a
convolutional network code (CNC), a delay may either be
introduced by the network, or inserted intentionally by the code
as part of the coding operation. As exemplified in [12], as long
as there is a positive delay along every cycle in the network,
data propagation in a causal manner can be assured over each
cycle by a CNC.
In order to guarantee causality, a CNC constructed by
existing algorithms, such as the deterministic one [3] and the
random one [5], is optimal (in the sense that achieves
asymptotically optimal data transmission rate) with respect to a
certain delay pattern. In this paper, after reviewing some
fundamentals on convolutional network coding in Section II,
we introduce a new class of CNCs, called delay invariant
CNCs (DI-CNCs), in a multicast network in Section III. A DICNC is optimal in the presence of any delays in the network. In
This work is supported by AoE grant E-02/08 from the University Grants
Committee of the Hong Kong Special Administration Region, China.
this way, the code can be constructed without the knowledge of
network delays. Moreover, once it is deployed into the network,
its optimality will not be affected by any delay changes brought
about by inappropriate synchronization or other issues. Even
though there are infinitely many (integer) delay patterns, we
show that every field-based optimal linear network code
actually qualifies as a DI-CNC. Moreover, we shall further
show that a DI-CNC exists over every symbol field, and
random coding actually suffices to construct one with high
probability. On the other hand, in Section IV, when the symbol
field is no smaller than the number of receivers, a variation of
the technique in [11] will be proposed to adapt the acyclic
algorithm of [7] for the deterministic construction of a DI-CNC
with scalar coding operations.
II.
FUNDAMENTALS ON CONVOLUTIONAL NETWORK
CODING
Conventions. We shall adopt the following convention unless
otherwise specified.
 A network means a finite directed multi-graph that
contains a unique node s with no incoming edges. This
node is called the source. No edge loops around a node.
The edge set is denoted by E. Every edge represents a
noiseless transmission channel of unit capacity.
 For every node v, denote by In(v) and Out(v), respectively,
the sets of its incoming and outgoing edges. An ordered
pair (d, e) of edges is called an adjacent pair when there is
a node v such that d  In(v) and e  Out(v).
 Outgoing edges from s are called data-generating edges.
Abbreviate |Out(s)| as , which represents the (fixed) data
generating rate from the source. Assume an ordering on
edges in E led by data-generating edges.
 A sink means a non-source node v to which there are 
edge-disjoint paths from s. The number of sinks in the
network will be denoted by .
 Let 𝔽 denote a finite field, which represents the symbol
alphabet, and 𝔽q the finite field with q elements. Let P be a
PID, which represents the general ensemble of data units.
In conventional network coding, P = 𝔽 and in
convolutional network coding, P = 𝔽[(D)].
 In a network, a P-linear network code (P-LNC), denoted
by (kd,e), assigns a coding coefficient kd,e  P to every pair
(d, e) of edges such that kd,e = 0 when (d, e) is not an
adjacent pair. An 𝔽-CNC means an 𝔽[(D)]-LNC (kd,e) with
kd,e  𝔽[D]  𝔽[(D)].
e1
1 1
e8
1
1
e7
u
1
e4
1
e6
e5
0
convolutional multicast is not necessarily optimal anymore. For
instance, consider the network depicted in Figure 3, and a delay
function t which assigns delay one to (e5, e3), (e8, e4), (e12, e6),
(e13, e8), (e9, e10), and delay zero to all others. If the efficient
algorithm of [3] is adopted to construct a t-causal 𝔽 convolutional multicast (kd,eDt(d,e)), a possible resulting code is
prescribed in Figure 3 by nonzero kd,e. In particular, the coding
1
𝐷 T
vectors for e4 and e12 are, respectively, (1−𝐷
and
1−𝐷)
1
1 T
(1+1−𝐷 1−𝐷) . However, if the delay function t is changed to
assign delay one to (e6, e7) and delay zero to (e8, e4), then the
new t-causal 𝔽-CNC (kd,eDt(d,e)) is no longer a convolutional
multicast, because the coding vectors for e4 and e12 become
1
1 T
identical, i.e., (1−𝐷
1−𝐷) , and thus sink t1 cannot reconstruct
both data streams. Therefore, under the new function t, the
code (kd,e) has to be redesigned in order to make it optimal.
Hence, we propose a type of CNC that is optimal with respect
to an arbitrary delay function.
e2
v
0
e3
e10
1
1
1
e9
1
Figure 1. An 𝔽2-CNC in the Shuttle Network [10] is prescribed by coding
coefficients kd,e. It is not normal, because there does not exist a set of coding
vectors for (kd,e). However, given an arbitrary delay function t, the 𝔽2-CNC
(kd,eDt(d,e)) becomes normal. For instance, if t assigns delay one to (e7, e3), (e9,
e5), (e10, e4), and delay zero to all others, the coding vectors of the t-causal
code (kd,eDt(d,e)) are fe1 = fe3 = (1 0)T, fe2 = fe10 = (0 1)T, fe4 = fe5 = fe6 =
( 11D
D T
,
1 D
)
fe7 = fe8 = fe9 = ( 11D
1 T
.
1 D
)
Consequently, it qualifies as a
Definition 4. An 𝔽-CNC (kd,e) is called a delay invariant 𝔽CNC (DI- 𝔽 -CNC) if for any delay function t, the code
(kd,eDt(d,e)) is a t-causal 𝔽-convolutional multicast.
delay invariant 𝔽2-CNC.
As an example, the 𝔽2-CNC depicted in Figure 1 is a DI-𝔽2CNC. One of the merits of DI-CNC is that the code design is
independent of delay functions. One possible design of a DICNC is to construct an 𝔽-linear multicast.
A P-LNC is said to be normal if it determines a unique set
of coding vectors. Normality of a LNC is a prerequisite for the
notion of data propagation via the code. In a network with
cycles, not every P-LNC is normal. A sufficient condition
stated in [11] for normality of a P-LNC (kd,e) is that det(I|E| 
[kd,e]d,eOut(s)) is a unit in P, where [kd,e]d,eOut(s) is a square
matrix with rows and columns indexed by E\Out(s). The
conditions for a CNC to be normal are discussed in [10] and [2].
In particular, a causal CNC is normal.
Proposition 5. Every 𝔽-linear multicast is a DI-𝔽-CNC.
Proof. Let (kd,e) be an 𝔽-linear multicast with coding vectors
fe. Given a delay function t, the t-causal 𝔽-CNC (kd,eDt(d,e)) is
normal by Proposition 2. Let fe be coding vectors determined
by (kd,eDt(d,e)). Define a homomorphism  that projects 𝔽[(D)]
onto 𝔽 by mapping D to 1. Applying componentwise, 
extends to a mapping from 𝔽[(D)] to 𝔽. The code (kd,eDt(d,e))
is an 𝔽-convolutional multicast because for every sink v,
Definition 1. A delay function t in a network assigns a nonnegative integer to every adjacent pair (d, e) such that, along
every cycle, there is at least one pair (d, e) with t(d, e) > 0. An
𝔽-CNC is said to be t-causal if the coding coefficient for every
adjacent pair (d, e) is divisible by zt(d, e).
rank 𝔽[(𝐷)] (fe:eIn(v))≥rank 𝔽 ((fe): eIn(v))
= rank 𝔽 (fe: eIn(v)) = .
In an acyclic network, every 𝔽-LNC is naturally a t-causal
𝔽-CNC with scalar coding coefficients in 𝔽[D], where t is a
zero function.
Even though a source always generates a time sequence of
messages to be propagated over a network, when the network is
acyclic, a LNC is always assumed to ignore the issue of data
communication delay and deal with each single message
individually. One reason is that upstream-to-downstream
ordering of nodes enables buffering, pipelining, and resulting
concerted synchronization among all nodes so that the
encoding and transmission of a message is independent of
sequential messages. This facilitates the model of zero delay
function. From another perspective justified by Proposition 5,
even if nonzero network delays are taken into account, the
design of a field-based optimal LNC still suffices to induce an
optimal LNC in the combined space-time domain.
Proposition 5 guarantees the existence of a DI-𝔽-CNC when
𝔽 is sufficiently large. In fact, we can say more.
Proposition 2 ([10]). A causal 𝔽-CNC is normal.
Example. For succinctness, all illustrations of networks in the
paper will omit the source node s. Figure 1 depicts the coding
coefficients (kd,e) of an 𝔽2-CNC in the Shuttle Network [10].
The code is not normal. However, given an arbitrary delay
function t, the t-causal 𝔽2-CNC (kd,eDt(d,e)) becomes normal.
Definition 3. A normal P-LNC with the coding vectors fe is
called a P-linear multicast when rankP(fe: eIn(v)) =  for
each sink v. A linear multicast is optimal in the sense that all
sinks can get enough information to recover the source
message.
III.
∎
Theorem 6. In any network, there exists a DI-𝔽-CNC, with
every polynomial coding coefficient of degree at most
O(|E|2logp|E|), where p is the characteristic of 𝔽.
Proof. The key is to establish a homomorphism from 𝔽[D]
onto an appropriate extension field 𝔽 of 𝔽 such that infinitely
many delays can be mapped to a finite number of values in 𝔽.
DELAY INVARIANT CONVOLUTIONAL NETWORK CODES
A causal 𝔽-convolutional multicast is always affiliated with
some delay function t. Therefore, the function t is always
known in advance for code construction. However, when t is
changed due to some unexpected reason, the already deployed
2
e1
1
t1
e6
1
e5
c(1)
1
e7
1
ad
0
e8
e9
0
a(3)
1 T
for e4
1 D
)
(a)
sink t1 cannot reconstruct both data streams.
Corollary 7. Consider an 𝔽-CNC with coding coefficients kd,e
independently and uniformly selected from those polynomials
2
in 𝔽[D] with degrees smaller than d. When pd > (2(p+1)d)|E| ,
where p is the characteristic of 𝔽, the probability for the code
𝛿(2(𝑝+1)𝑑)|𝐸|
𝑝𝑑
) .
∎
In comparison, random coding [5] constructs a causal 𝔽 qconvolutional multicast with respect to a delay function t at
probability at least (1 −
IV.
𝛿
𝑞𝑑
)
|𝐸|
d(3)
c3
d3
u4
(b)
Algorithm. Given a network  , the associated acyclic
network  consists of nodes in five layers, which are labeled
0 to 4 from upstream to downstream. The notation for a node
carries a subscript indicating the layer. All edges are between
adjacent layers.

Layer 0 consists of just the source node.
 Corresponding to each edge e  E\Out(s), there is a layer-1
node e1 and an edge e(1) from s to e1. Thus there are |E|– 
data-generating edges.
 Corresponding to each edge e  E, there is a node e2.
When e  E\Out(s), there is also an edge e(2) from e1 to e2.
Corresponding to every adjacent pair (d, e) in , there is
an edge de from e1 to d2.
 Corresponding to each edge e  E, there is a node e3 as
well as an edge e(3) from e2 to e3.
 Corresponding to each sink v in , there is a node v4.
Arbitrarily take  edge-disjoint paths in  that lead from
data-generating edges to v. For every e  E, install an edge
from e3 to v4 unless e  In(v) and is an edge on these paths.
2 |𝐸|
Proof. See Appendix.
d2
Figure 2. (a) The given network  contains a cycle. (b) The associated
acyclic network  in which the bottom layer nodes qualify as sinks. Every
𝔽-linear multicast on  subject to (1) induces a delay invariant
convolutional multicast on  via 0.
It then suffices to show the existence of a certain 𝔽 -LNC
which can induce a DI-𝔽-CNC. See Appendix for details.
∎
to be a DI-CNC is at least (1 −
b3
d(2)
cd dc
c2
c(3)
b(3)
v4
and e12, such that
c(2)
b2
a3
Figure 3. Under the delay function t that assigns delay one to (e5, e3), (e8, e4),
(e12, e6), (e13, e8), (e9, e10), and delay zero to all others, the algorithm of [3]
can construct an 𝔽-CNC (kd,e), which is depicted by nonzero coding
coefficients. The code (kd,eDt(d,e)) is a t-causal convolutional multicast.
Nevertheless, (kd,e) is not a DI-CNC. This is because when t is changed to
assign delay one to (e6, e7) and delay zero to (e8, e4), the new t-causal 𝔽-CNC
( 11D
bc
a2
0
1D
1 1D
1 0
(kd,eDt(d,e)) has identical coding vectors
d1
1
e13
10 1 10
1
d(1)
c1
e10
e11
e3
e2
1
0
e12
e4
1
1
0
when qd > .
EFFICIENT DETERMINISTIC CODE CONSTRUCTION
Although DI-CNCs can be generated by random coding,
coding coefficients have to be chosen from a sufficiently large
set of polynomials so as to guarantee high successful
construction probability. Thus, encoding complexity may
become very high. In this section, we shall demonstrate that
when |𝔽| ≥ , which is the number of sinks in a network , the
deterministic acyclic algorithm of [7] suffices to construct a
DI-𝔽-CNC with scalar coding coefficients, such that the coding
complexity becomes much lower:
 When  is acyclic, the algorithm of [7] directly constructs
an 𝔽-linear multicast, which is naturally a DI-𝔽-CNC by
Proposition 5. The computational complexity of the
algorithm is O(|E|(+ )).
 When  contains cycles, we can associate it with an
acyclic network  of  sinks via the algorithm below. An
𝔽-linear multicast subject to a straightforward condition (1)
on  can then be constructed by the algorithm of [7],
which in turn induces a DI-𝔽-CNC on  via (2). It takes
O(|E|3) to construct a DI-𝔽-CNC.
The association of  with  is illustrated by Figure 2. The
reference [11] proved that every layer-4 node in  is a sink.
Moreover, it justified that the algorithm of [7] is able to
construct an 𝔽-linear multicast on  subject to the following
(1) The coding coefficient is either 0 or 1 for every adjacent
pair in the form of (e(1), x). The coding coefficient is 1 for
every adjacent pair in the form of (e(1), e(2)) or (e(2), e(3)).
Proposition 8. Every 𝔽-linear multicast (kx,y) on  subject to
(1) induces a DI-𝔽-CNC (kd,e) on  via

kd ,e   k de,d 

( 3)
Proof. Let t be an arbitrary delay function on , and t a
delay function on  such that t(x, y) = t(d, e) if x = de and y
3
= d(3) for some adjacent pair (d, e) in . Else, t(x, y) = 0.
According to Proposition 5, the code (kx,yDt(x,y)) is a t-causal
𝔽-convolutional multicast. Let fe denote the coding vector of
edge e(3) for the code (kx,yDt(x,y)) on . Note that [fe]eE\Out(s) =
[kd,e]d,eIn(s). Since the 𝔽 -CNC (kd,eDt(d,e)) on  is tcausal, it is also normal by Proposition 2, which implies
det([kd,e]d,eIn(s)) ≠ 0. As a result, rank 𝔽[(𝐷)] (fe: e 
E\Out(s)) = , and hence Theorem 20 in [11] may be applied
to prove (kd,eDt(d,e)) in  to be a t-causal 𝔽 -convolutional
multicast on .
∎
[6]
[7]
[8]
[9]
[10]
[11]
V.
SUMMARY
[12]
The coding coefficients kd,e of a convolutional network code
(CNC) are always selected to be polynomials over a symbol
field. When they are implemented in the network, the effective
coding coefficients are in fact kd,eDt(d,e), where t is a delay
function on adjacent pairs of edges modeling the combination
of any data propagation delays. In this paper, we introduce the
concept of delay invariant CNCs (DI-CNCs), which are
optimal with respect to any delays. In particular, a CNC (kd,e) is
said to be delay invariant if associated with any delay function
t, the resultant CNC (kd,eDt(d,e)) is always optimal, i.e., a
convolutional multicast. The main result in this paper shows
the existence of a DI-CNC over any symbol field, and a
random coding technique that can produce a DI-CNC with high
probability when coding coefficients are selected from
sufficiently many polynomials. Moreover, when the symbol
field is no smaller than the number of sinks, a variation of the
technique in [11] can adapt the acyclic algorithm of [7] for the
construction of a DI-CNC with scalar coding coefficients. The
DI-CNC constructed by the deterministic algorithm thus has
much lower encoding complexities compared with the one
generated by random coding. However, even in an acyclic
network, it is still unclear whether there is a polynomial time
algorithm for DI-CNC construction over a smaller symbol
field, for instance the binary field. Any investigation along this
line is very intriguing.
[13]
ACKNOWLEDGMENT
The authors appreciate helpful discussions with Pak Hou
Che, Chung Chan and Xunrui Yin.
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Applying componentwise, the homomorphism  also extends
to a mapping from matrices over 𝔽 [D][] to matrices over
𝔽22∙3𝑚 []. Because  has order 3𝑚+1 , for each sink v,
APPENDIX. PROOF OF THEOREM 6 AND COROLLARY 7
Convention. For a normal P-LNC (kd,e), we adopt the
abbreviations
2
|({Mv(t): t is a delay function})| < 3(𝑚+1)|𝐸| .
Ak = [kd,e]dIn(s),eIn(s), Bk =  [kd,e]d,eIn(s).
Then, the coding vector fe for e  E\In(s) can be represented by
fe =
𝐴𝑘  𝐴𝑑𝑗(𝐵𝑘 )  𝑢𝑒
det(𝐵𝑘 )
Consequently, the highest degree of every indeterminate in the
2
polynomial t:delay function(det(Mv(t))) is less than 3(𝑚+1)|𝐸| .
Therefore, when a subset F  𝔽22∙3𝑚 has more than
2
3(𝑚+1)|𝐸| elements, there exist values kd,e  F such that the
evaluation of
,
where Adj() means the adjugate of a square matrix, and ue an
(|E|)-dim column vector indexed by E\In(s) such that the
entry labeled by e is 1 and all others are 0.
For a subfield 𝔽p of 𝔽 and an element   𝔽, denote by 𝔽p()
the smallest subfield of 𝔽 containing 𝔽p and .
∎
v: sinkt: delay function(det(Mv(t)))
at xd,e = kd,e is not 0  𝔽22∙3𝑚 . Since
|𝔽22∙3𝑚 | 22|E| log2 |E| | E |2|E| | E |2( 1)|E|
2
Now associate every adjacent pair (d, e) with an
indeterminate xd,e. Let 𝔽[D][] denote the polynomial ring in
these indeterminates over 𝔽[D]. Write xd,e = 0 when (d, e) is not
an adjacent pair. For every sink v, select  edge-disjoint paths
from data-generating edges to  distinct edges belonging to
In(v). Denote by Pv the juxtaposition of ue, where e is among
the said  distinct edges in In(v). Moreover, given a delay
function t, let Mv(t) denote the |E||E| matrix
 [ xd ,e D t ( d ,e ) ]dIn( s ),eIn(s )

t ( d ,e )
]d ,eIn( s )
 I |E| [ xd ,e D
2
2
| E |2|E|  9 log2 | E ||E| = (| E |2 9 log2 | E |)|E|
2
2
2
 3( m1)|E|
2
the choice of such F is possible. Let P denote the set of those
polynomials in 𝔽[D] that have degrees smaller than 23m =
O(|E|2log2|E|). Since 𝔽2() = 𝔽22∙3𝑚 , the minimal polynomial
of  over 𝔽 has degree 23m. Therefore, (P) = 𝔽22∙3𝑚 , and the
𝔽2-CNC with coding coefficients kd,e = P –1(kd,e) abides by
(3). In order to show Corollary 7 under 𝔽 = 𝔽2, redefine m to be
the smallest integer such that 23m ≥ d > 23m –1. Let P denote
the set of those polynomials in 𝔽[D] that have degrees smaller
than d. Note that the polynomial v:sinkt:delayfunction
2
(det(Mv(t))) over 𝔽22∙3𝑚 is of degree less than |E|3(𝑚+1)|𝐸| ,
and the highest degree of every indeterminate in it is less than
2
 3(𝑚+1)|𝐸| . Thus, Lemma 4 in [5] asserts that when
indeterminates xd,e are independently and uniformly assigned to
be values kd,e in (P)  𝔽22∙3𝑚 ,
0
.
Pv 
In order to qualify an 𝔽-CNC (kd,e) as a DI-CNC, it suffices to
show that
(3) Given any delay function t, the evaluation of det(Mv(t)) at
xd,e = kd,e is not 0  𝔽[D] for all sinks v.
To justify this, consider the t-causal 𝔽-CNC (hd,e), where hd,e =
kd,eDt(d,e). By Proposition 2, the code is normal, and hence
det(Bh) ≠ 0 and det(Adj(Bh)) ≠ 0. Moreover, similar to the proof
of Lemma 1 in [5], because
Pr(v:sinkt:delayfunction(det(Mv(t)) ≠ 0) > (1 −
 Ah 0   Adj( Bh ) Pv Adj( Bh )
 B P    det( Bh ) I
0 
  h v  

𝛿∙3(𝑚+1)|𝐸|
2𝑑
2 |𝐸|
)
Consequently, when coding coefficients of an 𝔽 2-CNC are
independently and uniformly chosen from P –1(kd,e), the
probability for the code to be delay invariant is at least
 A  Adj ( Bh )  Pv Ah  Adj ( Bh ) 
  h
, 
0
det( Bh ) I |E| 

(1 −
 A 0 
if det   h
  ≠0, then det(AhAdj(Bh)Pv)≠0, and thus
  Bh Pv  
𝛿(5𝑑)|𝐸|
2𝑑
2 |𝐸|
)
< (1 −
𝛿∙3(𝑚+1)|𝐸|
2𝑑
2 |𝐸|
)
(Since 5d > 3m+1.)
Take 𝔽 = 𝔽p, where p is an odd prime number, and let m be
the integer such that 2m ≥ |E|2logp|E| ≥ 2m–1, where  is a
2
constant subject to | E | 2  1/|E| 2 ( p 1)log p | E |. Define a
rank𝔽[(D)](fe: eIn(v)) = rank𝔽[(D)](AhAdj(Bh)[ue]eIn(v)) = .
homomorphism : 𝔽[D]  𝔽𝑝2𝑚 that fixes 𝔽 and maps D to ,
where  is of order 2m(p + (–1)(p+1)/2) and 𝔽p() = 𝔽𝑝2𝑚 . Lemma
10 asserts the existence of such . By an argument similar to
the case when 𝔽 = 𝔽2, we can prove Theorem 6 and Corollary 7
under 𝔽 = 𝔽p.
Take 𝔽 = 𝔽q, where q is a positive integer power of a prime
p. Let D and D, respectively, represent the unit time to carry
First take 𝔽 = 𝔽 2 and let m be the integer such that 3m ≥
|E|2log2|E| > 3m–1, where  is a constant subject to
| E |2( 1)  1/|E| 9 log2 | E |. Define a homomorphism :
𝔽[D][]𝔽22∙3𝑚 [] that fixes 𝔽 as well as indeterminates xd,e,
and maps D to , where  is an element in 𝔽22∙3𝑚 of order
3𝑚+1 and 𝔽2() = 𝔽22∙3𝑚 . Lemma 9 asserts the existence of .
2
5
one data symbol in 𝔽p and 𝔽q. Define a homomorphism  from
𝔽 p[(D)] into 𝔽 q[(D)] via fixing 𝔽 p and mapping D to D.
Because the mapping  is one-to-one, given a DI-𝔽p-CNC (kd,e)
and a delay function t, ((kd,eDt(d,e))) qualifies as an 𝔽 qconvolutional multicast too.
∎
24∙3
𝑖−1
+ 22∙3
𝑖−1
+ 1 ≡ 3 mod 9.
∎
Lemma 10. Let p be a prime of odd characteristic. In 𝔽𝑝2𝑚 , m
≥ 1, there exists an element  of order 2𝑚 (𝑝 + (−1)(𝑝+1)/2 )
such that 𝔽p() = 𝔽𝑝2𝑚 .
Proof. Let  be a primitive element in 𝔽𝑝2𝑚 . First,
Lemma 9. In 𝔽22∙3𝑚 , where m ≥ 0, there exists an element  of
order 3𝑚+1 such that 𝔽2() = 𝔽22∙3𝑚 .
Proof. Let  be a primitive element in 𝔽22∙3𝑚 . First, 3𝑚+1
𝑚
𝑚
divides 22∙3 − 1, because by Euler's totient theorem, 22∙3 ≡
𝑚
2∙3 −1)/3𝑚+1
1 mod 3𝑚+1 . Let  = (2
and 𝔽2() = 𝔽2𝑑 , where
either
𝑖
 𝑑 = 3 , 0  i  m, or
𝑖
 𝑑 = 2 ∙ 3 , 0  i  m.
𝑑
Since  has order 3𝑚+1 and 2 −1 = 1 , 3𝑚+1 | 2𝑑 − 1. Thus,
𝑖
d ≠ 3𝑖  0  i  m, because 23 − 1 ≡ 1 mod 3. Assume d =
𝑖
𝑚−1
2 ∙ 3𝑖 , 0  i  m. Because 22∙3 − 1 | 22∙3
− 1 for all i < m,
𝑚−1
in order to show i = m, it suffices to prove 3𝑚+1 ∤ 22∙3
−
1. When m = 1, it is trivially true that 3 is not divisible by 9.
𝑚−1
𝑚−2
𝑚−2
When m > 1, note that 22∙3
− 1 = (22∙3
− 1)(24∙3
+
𝑚−2
𝑚−2
22∙3
+ 1). By the induction hypothesis, 3𝑚 ∤ 22∙3
− 1.
𝑚−1
𝑚−2
𝑚−2
Therefore, if 3𝑚+1 | 22∙3
− 1 , then 24∙3
+ 22∙3
+1
would be divisible by 9. However, this is impossible because
𝑚
2𝑚−1 (𝑝2 − 1) | 𝑝 2 − 1, since
𝑚
𝑝2 − 1 = (𝑝 2
𝑚−1
+ 1)(𝑝 2
𝑚−2
Assume 𝑝 ≡ 1 mod 4. Let  = 
+ 1) … (𝑝 + 1)(𝑝 − 1).
𝑚
(𝑝2 −1)
2𝑚 (𝑝−1)
m. Since  has order 2𝑚 (𝑝 − 1) and 
𝑘
divides 𝑝 2 − 1, which implies
2𝑚 | (𝑝 2
𝑘−1
+ 1)(𝑝2
𝑘−2
and () = 𝔽
𝑘
𝑝2 −1
𝑝2
𝑘
,k
= 1, 2𝑚 (𝑝 − 1)
+ 1) … (𝑝 + 1).
𝑘
𝑖
Moreover, because 𝑝2 + 1 ≡ 2 mod 4, the factors 𝑝2 + 1, 0
 i < k, can be divisible by 2 but not by 4. Therefore, k = m.
The case when 𝑝 ≡ 3 mod 4 can be shown analogously
𝑚
(𝑝2 −1)
such that 𝔽() = 𝔽𝑝2𝑚 , where  = 2𝑚(𝑝+1) .
6
∎