ERROR-detecting codes and error-correcting codes work

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1 1 Convolutional-Coe-Specific CRC Coe Design Chung-Yu Lou, Stuent Member, IEEE, Babak Daneshra, Member, IEEE, an Richar D. Wesel, Senior Member, IEEE arxiv: v1 [cs.it] 9 Jun 2015 Abstract Cyclic reunancy check CRC coes check if a coewor is correctly receive. This paper presents an algorithm to esign CRC coes that are optimize for the coe-specific error behavior of a specifie feeforwar convolutional coe. The algorithm utilizes two istinct approaches to computing unetecte error probability of a CRC coe use with a specific convolutional coe. The first approach enumerates the error patterns of the convolutional coe an tests if each of them is etectable. The secon approach reuces complexity significantly by exploiting the equivalence of the unetecte error probability to the frame error rate of an equivalent catastrophic convolutional coe. The error events of the equivalent convolutional coe are exactly the unetectable errors for the original concatenation of CRC an convolutional coes. This simplifies the computation because error patterns o not nee to be iniviually checke for etectability. As an example, we optimize CRC coes for a commonly use 64-state convolutional coe for information length k=1024 emonstrating significant reuction in unetecte error probability compare to the existing CRC coes with the same egrees. For a fixe target unetecte error probability, the optimize CRC coes typically require 2 fewer bits. Inex Terms catastrophic coe, convolutional coe, cyclic reunancy check CRC coe, unetecte error probability. I. INTRODUCTION ERROR-etecting coes an error-correcting coes work together to guarantee a reliable link. The inner errorcorrecting coe tries to correct any errors cause by the channel. If the outer error-etecting coe etects any resiual errors, then the receiver will eclare a faile transmission. Unetecte errors result when an erroneously ecoe coewor of the inner coe has a message that is a vali coewor of the outer coe. This paper esigns cyclic reunancy check CRC coes for a given feeforwar convolutional coe such that the unetecte error probability is minimize. A. Backgroun an Previous Work A necessary conition for a goo joint esign of erroretecting an error-correcting coes, both using linear block coes, is provie in [1]. However, this conition is base on the minimum istances of the inner an outer coes an oes not consier the etaile coe structure. Most prior work on CRC esign ignores the inner coe structure by assuming that the CRC coe is essentially operating on a binary symmetric channel BSC. We refer to this as the BSC assumption. The BSC assumption oes not take C.-Y. Lou, B. Daneshra, an R. D. Wesel are with the Department of Electrical Engineering, University of California, Los Angeles, CA USA chungyulou@ucla.eu; babak@ee.ucla.eu; wesel@ee.ucla.eu. This material is base upon work supporte by the National Science Founation uner Grant Number Any opinions finings, an conclusions or recommenations expresse in this material are those of the authors an o not necessarily reflect the views of the National Science Founation. avantage of the fact that the CRC coe will only encounter error sequences that are vali coewors of the inner coe. The unetecte error probability of a CRC coe uner the BSC assumption was evaluate using the weight enumerator of its ual coe in [2]. Fast algorithms to calculate ominant weight spectrum an unetecte error probability of CRC coes uner the BSC assumption were presente in [3], [4]. In [5], Koopman an Chakravarty list all stanar an goo uner the BSC assumption CRC coes with up to 16 parity bits for information lengths up to 2048 bits. The authors recommen CRC coes given the specific target reunancy length an information length. For more than 16 CRC parity bits, it is ifficult to search all possibilities an fin the best CRC coes even uner the BSC assumption. Some classes of CRC coes with 24 an 32 bits were investigate uner the BSC assumption in [6] an an exhaustive search for 32-bit CRC coes uner the BSC assumption was performe later in [7]. Because all of these esigns ignore the inner coe by assuming a BSC, there is no guarantee of optimality when these CRC coes are use with a specific inner coe. A few papers o consier CRC an convolutional coes together. In [8], the CRC coe was jointly ecoe with the convolutional coe an use to etect the message length without much egraation of its error etection capability. In [9], CRC bits were puncture to reach higher coe rates. The authors notice that bursty bit errors cause an impact on the performance of the puncture CRC coe. Recently, [10] an [11] have consiere a CRC coe use for error correction jointly with convolutional an turbo coes, respectively. However, in these cases, the error etection capability of the CRC coe is egrae. Such unetecte error probability as well as false alarm probability were analyze in [12] uner the BSC assumption. The authors of [12] also moele the bursty error at the turbo ecoer output using a Gilbert-Elliott channel. B. Main Contributions We propose two methos to compute the unetecte error probability of a CRC coe concatenate with a feeforwar convolutional coe. The exclusion metho enumerates possible error patterns of the inner coe an exclues them one by one if they are etectable. The construction metho constructs a new convolutional coe whose error events correspon exactly to the unetectable error events of the original concatenation of CRC an convolutional coes. With these two methos as tools, we esign CRC coes for the most common 64- state convolutional coe for information length k = 1024 an compare with existing CRC coes, emonstrating the performance benefits of utilizing the inner coe structure.

2 2 CRC Encoer Convolutional Encoer QPSK Moulator m fx x f x r x q x p x q x p x c x Fig. 1. CRC Decoer Viterbi Decoer QPSK Demoulator AWGN Channel Block iagram of a system employing CRC an convolutional coes. This paper is organize as follows: Section II provies the system moel. Section III presents the exclusion an construction methos for computing the unetecte error probability of a CRC coe concatenate with an inner convolutional coe. Section IV escribes how these two methos can be use to esign a CRC coe to minimize the unetecte error probability for a specific feeforwar convolutional coe an information length. Section V applies this esign approach to the most common 64-state convolutional coe for information length k = Section VI conclues the paper. II. SYSTEM MODEL Fig. 1 shows the block iagram of our system moel employing a CRC coe concatenate with a convolutional coe in an aitive white Gaussian noise AWGN channel. The k-bit information sequence is expresse as a binary polynomial fx of egree smaller than or equal to k 1. The m parity bits are the remainer rx of x m fx ivie by the egree-m CRC generator polynomial px. Thus the n = k m-bit sequence escribe by x m fxrx is ivisible by px proucing the k-bit quotient qx an a remainer of zero. The CRC-encoe sequence can also be expresse as qxpx. Thus qx has a one-to-one relationship with fx. Note that qxpx is the result of processing the sequence x m qx qx an m trailing zeros by the CRC encoer circuit escribe by px. The transmitter uses a feeforwar, terminate, rate- 1 N convolutional coe having ν memories with generator polynomial cx = [c 1 x, c 2 x,, c N x]. The output qxpxcx of the convolutional encoer is sent to the AWGN channel using quarature phase-shift keying QPSK moulation. Because the CRC bits are ae at the en of the sequence, the highest egree term of fx is the bit that is first in time an first to enter the convolutional encoer. Consistent with this convention an in contrast to common representations, the highest egree terms of cx represent the most recent encoer input bits. Thus a convolutional encoer with the generator GD = [1 D 3 D 4, 1 D D 2 ] will have cx = [x 4 x 1, x 4 x 3 x 2 ]. The emoulate symbols are fe into a soft Viterbi ecoer. The CRC ecoer checks the n-bit sequence resulting from Viterbi ecoing. An unetecte error occurs when the receiver eclares error-free ecoing when the Viterbi ecoer ientifie an incorrect coewor. This work can be applie to feeback convolutional coes as well. Consier a feeback convolutional coe with generator polynomial cx/c FB x, where c FB x is its feeback connection polynomial. This feeback coe has the same set of coewors as a feeforwar coe with generator polynomial cx. Assume the Viterbi ecoer selects a wrong message tx at the feeforwar convolutional ecoer output. In most cases, the same receive signal is ecoe as txc FB x by the feeback convolutional ecoer. However, it is possible that the message ecoe by the feeback ecoer is not a multiple of c FB x. Such trellis eviation must happen uring the termination of the coewor an correspons to either a long error sequence with large coewor Hamming istance or a short error sequence occurring only at the en of the coewor. Either of these cases shoul not ominate the overall coewor error performance an thus we only consier message errors of the form txc FB x. Let the greatest common ivisor of px an c FB x be c gc x. The polynomial px ivies txc FB x if an only if px/c gc x ivies tx. Hence, the unetecte error probability of this CRC coe concatenate with the feeback convolutional coe is approximate by that obtaine by the CRC coe px/c gc x concatenate with the feeforwar convolutional coe cx an can be analyze using the methos presente in this paper. Furthermore, a smart choice when using a CRC coe concatenate with a feeback convolutional coe is to pick px an c FB x relatively prime. III. UNDETECTED ERROR PROBABILITY ANALYSIS Let ex be the polynomial of error bits in the Viterbiecoe message so that the ecoe n-bit sequence followe by ν zeros for termination is expresse as qxpxx ν ex. If ex 0 is ivisible by px, then this error is unetectable by the CRC ecoer. This section presents two methos, the exclusion metho an the construction metho, to calculate the probability that a non-zero error ex occurs that is unetectable. A. Exclusion Metho The exclusion metho enumerates the possible error patterns of the convolutional coe an exclues the patterns etectable by the CRC coe. The probability of the unexclue error patterns is the unetecte error probability. The exclusion metho filters out part of the istance spectrum of the convolutional coe through a ivisibility test to create the istance spectrum of the unetectable errors of the concatenate coe. 1 Unetectable Single Error: An error event occurs when the ecoe trellis path leaves the encoe trellis path once an rejoins it once. Let e,i x be the polynomial of message error bits associate with the i th error event that leas to a coewor istance from the transmitte coewor, where the range of i is later specifie in 1. Note that in this paper, the term istance always refers to the convolutional coe output Hamming istance. Let this error event have length l,i. Both e,i x an l,i are obtaine through computer search of the given convolutional coe. The first highest power term of e,i x is x l,i 1 an the last lowest power term is x ν because every error event starts with a one an ens with a one followe by ν consecutive zeros. If the receive ata frame contains only one error event, then the polynomial of message error bits can be expresse as ex = x g e,i x, where g [0, n ν l,i ] inicates the possible locations where this error event may appear. If e,i x

3 3 is ivisible by px, this error event, incluing all of its offsets g, will be unetectable. The union boun of such error probability is given by a P UD,1 Ipx e,i x = free i=1 max 0, n ν l,i 1} P, 1 Fig. 2. l 1 g 1 An illustration of two error events. l 2 g 2 where free is the free istance of the convolutional coe, a is the number of error events with output istance, the inicator function I returns one when e,i x is ivisible by px an zero otherwise, an P is the pairwise error probability of an error event with istance. The max operator ensures that the number of possible locations is always nonnegative even when l,i is large. Note that while 1 oes not explicitly use the generator polynomial of the convolutional coe, it oes implicitly epen on the generator polynomial because the generator polynomial etermines the vali trellis error events e,i x. The subscript 1 in P UD,1 means that this probability only inclues unetectable errors that are single error events. We call this type of error an unetectable single error. For a QPSK system operate in an AWGN channel, P can be compute using the tail probability function of stanar normal istribution, i.e. Gaussian Q-function, as [13] P = Q 2γ Q 2free γ e freeγ, 2 where γ = E s /N 0 is the signal-to-noise ratio SNR of a QPSK symbol, an E s an N 0 /2 enote the receive symbol energy an one-imensional noise variance, respectively. Note that the accuracy of 2 comes in part from a knowlege of free. A useful Q-function approximation when knowlege of free is not available is presente in [14]. To generalize 2 to a higher-orer quarature amplitue moulation QAM system with bit-interleave coe moulation using a ranom interleaver, one can multiply γ with a moulation-epenent factor to obtain an approximation. Details can be foun in [14] as well. To compute P UD,1 each error event e,i must first be ientifie as either ivisible by px or not. One approach is to truncate 1 at to get an approximation, in which case all error events with istance can be store an this set of error events can be teste for ivisibility by the CRC polynomial px. The choice of is base on the computational an storage capacity available to implement an efficient search such as [15]. The require memory size to store the error events is proportional to a = free i=1 l,i. The approximation of 1 can be quite tight if the probability of the terms with > is negligible. However, assuming that all error events with > are unetectable provies P UD,1 = 1 n a P a = free i=1 P Ipx e,i x max 0, n ν l,i 1} }, 3 where l,i for > is replace with ν 1 because the shortest error event has length ν 1. Note that 3 can be compute because the error pattern e,i x is only require for, an the istance spectrum a for > provie by the transfer function [16] can be use for the first term of 3 as in the frame error rate FER bouns of [17]. In aition to the unetectable single error events iscusse above, an unetectable error coul consist of two or more error events, even though each of the error events itself is etectable. We will first iscuss the case with two error events, an then generalize it to multiple error events. 2 Unetectable Double Error: A ouble error involves two error events e 1,i 1 x an e 2,i 2 x with respective lengths l 1,i 1 an l 2,i 2. To simplify notation, for u 1, 2} let e u x an l u refer to e u,i u x an l u,i u, respectively. In a ata frame with two error events, the polynomial of error bits in the message can be expresse as ex = x g1g2l2 e 1 x x g2 e 2 x, where the exponents of two x s tell the locations of the two error events. Furthermore, g 1 0 represents the interval of symbols gap between two error events an satisfies g 1 g 2 l 1 l 2 n ν. If x g1l2 e 1 x e 2 x is ivisible by px, the error is an unetectable ouble error. Its length is l 1 l 2 g 1 an its offset is g 2. An upper boun of the probability of an unetectable ouble error occurring in the coewor is given by P UD,2 a 1 a 2 1= free 2= free i 1=1 i 2=1 nν l 1 l 2 g 1=0 I px x g1l2 e 1 x e 2 x P 1 2 n ν l 1 l 2 g The istance of the ouble error event is simply the sum of the iniviual istances because the error events are completely separate as shown in Fig. 2. Two error events that overlap are simply a longer single error event, which was treate in Section III-A1. Computation of 4 exactly is problematic because it requires infinite search epth. Replacing the l 1 an l 2 of largeistance terms in 4 with ν 1 yiels a more computationfrienly upper boun similar to 3 as follows: P UD,2 a 1 a 2 1, 2 D,2 i 1=1 i 2=1 nν l 1 l 2 g 1=0 I px x g1l2 e 1 x e 2 x n ν l 1 l 2 g 1 1 n ν n ν 1 2 1, 2 / D,2 1 P 1 2 a 1 a 2 P 1 2, 5 where D,2 = 1, 2 1, 2 free, 1 2 }.

4 4 Because computational complexity limits the single error event istance we can search, it is feasible to replace in D,2 with free. This is not particularly helpful because we have alreay assume errors with istance > are negligible or unetectable uring the calculation of unetectable single errors. We can also replace all l 1 an l 2 in 5 with ν 1 an provie another upper boun which oes not require any length information. As escribe in Appenix A, the number of g 1 values at which to check the ivisibility of x g1l2 e 1 xe 2 x by px can be significantly reuce from n ν l 1 l Total Unetecte Error Probability: In general, s error events coul possibly form an unetectable s-tuple error, whether each of them is etectable or not. The error bits in the message can be expresse as s s ex = x gvlv x gu e u x. 6 v=u1 This combine error will be unetectable if ex is ivisible by px. Therefore, the probability of unetectable s-tuple errors P UD,s can be approximate or boune in the same way as 5. For simpler notation, efine sets } s D,s = 1,, s u free u, u I s = i 1,, i s i u [1, a u ] u [1, s]} s G s = g 1,, g s g s1 u 0 u, g u n ν l u }. Note that G s is etermine by all u an i u, an I s is etermine by all u. Since an unetectable error may consist of any number of error events, the probability of having an unetectable error in the coewor P UD is upper boune by P UD,s. Using the computation-frienly boun of each term such as 3 an 5, we obtain s P UD P >,s u s=2 1,, s D,s P Ipx ex i 1,,i s I s g 1,,g s 1 G s 1 } s s 1 n ν l u g u 1 P1 1 D,1 } Ipx e 1 x max 0, n ν l 1 1}, 7 i 1 I 1 where the composition of ex epens on the number of error events s as in 6 an P >,s is the sum of probability of all s-tuple errors whose istances are greater than. The probability sum of all large-istance s-tuple errors is n ν sν s s P >,s = a u P u, 8 s 1,, s / D,s where the combinatorial number represents the number of ways to have s length-ν 1 error events in a length-n ν sequence. Using 2 P >,1 can be upper boune by P >,1 Q 2free γ e freeγ P n a 1 e 1γ, 9 1 D,1 where P is efine as P ntd, L D=e γ,l=1 using the transfer function [16] a TD, L = D L l,i. 10 = free i=1 Therefore, the sum of P >,s terms can be upper boune by s s P >,s a u P u 1,, s / D,s ns s! 1,, s / D,s n s Q 2free γ e freeγ s! s } a u e uγ = Q 2free γ e freeγ 1 s! P s ns s } a u e uγ s! 1,, s D,s = Q e P 2free γ e freeγ 1 n s s! 1,, s D,s 11a 11b s a u e }. uγ 11c The boun of Gaussian Q-function 2 is use in 11a, an the transfer function is use to evaluate the sum of all s-tuple errors in 11b. Using 7 an 11c, a boun of P UD can be calculate. In fact, when an unetectable error occurs in a coewor, the receiver may still etect an error if a etectable error happens somewhere else in the coewor. Therefore, P UD, which is the probability of having an unetectable error in the coewor, is an upper boun of the unetecte error probability. When channel error rate is low, having a etecte error occur along with the unetecte error is a rare event so that our boun will be tight. Due to the limitation of searchable epth of error events, the exclusion metho is only useful when unetectable errors with istances ominate. However, this requirement coul be violate by a powerful CRC coe that etects the short-istance errors. The next subsection introuces the construction metho, which allows the search epth to increase to istance ˆ >. B. Construction Metho The construction metho utilizes the fact that all unetectable errors at the CRC ecoer input are multiples of the

5 5 m x m x q x q x p x x x 2 1 CRC Coe State c x x x Original Convolutional Coe State Equivalent Convolutional Coe State c eq x x x x x x x Fig. 3. An example of CRC coe, original convolutional coe, an equivalent convolutional coe. TABLE I AN EXAMPLE OF STATE TYPES WITH px = x 2 x 1 AND ν = 2. State Equivalent Coe Original Coe Time Type State Input State Input 0 S Z S N S N S D S D S N S N S N S Z CRC generator px. This metho constructs an equivalent convolutional encoer c eq x = pxcx to isolate these errors. The set of non-zero coewors of c eq x is exactly the set of erroneous coewors given an all-zeros transmission that lea to unetectable errors for the concatenation of the CRC generator polynomial an the original convolutional encoer. Thus the probability of unetectable error is exactly the FER of c eq x. Fig. 3 shows an example of how c eq x is constructe from cx an px, where q x with m ν trailing zeros is the input that generates the unetectable erroneous coewor. The input/output behavior of c eq x is exactly the same as the concatenation of px an cx. For px with m memory elements an cx with ν memory elements, c eq x has mν memory elements. At a given time, the state of the original encoer cx can be inferre from the state of c eq x because the state of c eq x contains the last m ν inputs to px which are sufficient to compute the last ν outputs of px, which exactly comprise the state of cx. 1 States of the Equivalent Encoer: Define the all-zero state S Z to be the state where all memory elements of the equivalent encoer are zero. When the equivalent encoer is in S Z, then the original encoer is also in its zero state. Avoiing trivial cases by assuming that the x m 1 an x 0 coefficients of px are 1, there are 2 m 1 equivalent encoer states in aition to S Z that correspon to the all-zero state of the original encoer. To see this, consier the top iagram in Fig. 3 in which the equivalent encoer state is shown as the state of the CRC encoer extene with ν aitional memories. Note that whatever state the equivalent encoer is in, there is a sequence of ν bits that will prouce ν zeros at the output of the CRC encoer that will rive the original encoer state to zero. The specific set of ν bits is a function of the m-bits of state in the CRC encoer. Thus for any m-bits pattern there is a corresponing m ν-bit state of the equivalent encoer that correspons to the original encoer being in the zero state. We call the 2 m 1 non-zero equivalent encoer states for which the original encoer state is zero etectable-zero states, forming a set S D, because they correspon to the termination of a etectable error event in the original encoer. The remaining 2 mν 2 m states of the equivalent coe are calle non-zero states, forming a set S N, because the corresponing states of the original coe are not zero. To terminate an error event in the original encoer, the trellis of the equivalent coe transitions from a state in S N to S Z or to a state in S D. If the transition is to S D the cumulative errors are etectable because the portion of q xpx till this moment is not ivisible by px. If the transition is to S D, the cumulative errors are etectable because the portion of q xpx till this moment is ivisible by px. Table I shows an example using the set-up shown in Fig. 3 with px = x 2 x 1, q x = x 3 x 2 1, an ν = 2. The original encoer cx oes not nee to be specifie for the results in Table I because any feeforwar encoer will prouce the same state sequence. The zero state of the original coe is visite at time 0, 3, 4, an 8. At time 0, the state begins from S Z. At time 3, the first cx error event ens. This error event is etectable if the coewor ens at time 3 because the input to the original coe, i.e. x 2, is not ivisible by px. The state remains in S D at time 4. At time 5 a secon cx error event begins. At time 8, the c eq x state returns to S Z, which is not only the en of the secon cx error event but also the en of an unetectable ouble error. Note that c eq x is catastrophic because its generator polynomials have a common factor px. The catastrophic behavior is expresse through the zero istance loops that occur as the equivalent encoer traverses a sequence of states in S D while the original convolutional encoer stays in the zero state. Thus time spent in S D between cx error events can lengthen an unetectable error without increasing its istance. 2 Error Events in the Equivalent Encoer: Since the allzero coewor is assume to be sent, the correct path remains in S Z forever. An error event in the equivalent encoer occurs if the trellis path leaves S Z an returns S Z without any visits to S Z in between. We will classify these error events accoring to the number of times it enters S D from S N uring the eviation. If a trellis path enters S D from S N s 1 times, then it is classifie as an unetectable s-tuple error. In an unetectable s-tuple error, there are exactly s segments of consecutive transitions between states in S N, which correspon to s error events of the original encoer. These segments are separate by segments of consecutive transitions between states in S D. Fig. 4 illustrates an unetectable triple error in a system with ν = 2 an m = 2. The three error events of the original coe are separate by visits to S D. The path can leave S D right after entering it as shown between the first an the secon error events; the path can also stay in S D for a while an then leave S D from a state ifferent from where it enters S D as shown between the secon an the thir error events. Note that S Z is

6 6 0000,0001,0010,0101,1011,0111,1110,1101,1011,0110,1101,1011,0110,1100,1000,0000 0,1,2,5,11,7,14,13,11,6,13,11,6,12,8,0 1 st error 2 n error 3 r error Fig. 4. A trellis iagram of an equivalent convolutional coe with ν = 2 an m = 2 having an unetectable triple error. The states are reorere such that the all-zero state is at the top an the etectable-zero states are at the bottom. never irectly connecte to any state of S D. The probability of unetectable single errors is boune in a similar way as 3 by P UD,1 P > ˆ,1 a ZZ D ˆ,1 i=1 S Z S N S D max 0, n ν l ZZ,i 1 } P, 12 where a ZZ is the number of error events with istance starting from S Z an ening at S Z while never traversing a state in S D. The length l,i ZZ is the length of the ith error event counte in a ZZ. Note that this expression requires the[ istance spectrum a ZZ an length spectrum l,i ZZ for free, ˆ ] obtaine using computer search. Unlike 3, 12 oes not nee to check the ivisibility of error events. Hence, there is no nee to store the error patterns anymore but only their istances an lengths. Consequently the search epth ˆ can go beyon. Let S D i be the i th state in S D, where i [1, 2 m 1]. Define i as the subset of S D compose of all states connecte with S D i through a path that only inclues states in S D. Appenix B shows that i an the x-cyclotomic coset moulo px iscusse in Appenix A are equivalent. Thus if px is a primitive polynomial, all sets i are ientical an i = 2 m 1 is maximize. This may lea to fewer unetectable ouble errors as shown below an is esirable in the esign of a CRC coe. Similar to 5, the probability of unetectable ouble errors can be boune by P UD,2 P > ˆ,2 where 1, 2 D ˆ,2 nν l min / φ t 1=0 2 m 1 P 1 2 φ=1 a ZD,φ 1 a DZ,ψ 2 S D ψ φ i 1=1 i 2=1 n ν l min φ t 1 1, 13 l min = l ZD,φ 1,i 1 l DZ,ψ 2,i 2 δ φ,ψ 14 is the shortest possible length of the unetectable ouble error specifie by 1, 2, φ, ψ, i 1, i ,0 0001,1 0010,2 0011,3 0100,4 0101,5 0111,7 1000,8 1001,9 1010, , , , ,6 1011, ,13 In 13, φ is the inex of the state at which the trellis enters S D at the en of the first error event, an ψ is the inex of the state at which the trellis leaves S D at the beginning of the secon error event. In 14, δ φ,ψ < φ is the number of hops require to go from S D φ to S D ψ without leaving S D for S D ψ φ, where δ φ,ψ = 0 if φ = ψ. The number of error events starting at S Z an ening at S D φ with istance 1 is a ZD,φ th 1, an the i 1 error event of them has length l ZD,φ 1,i 1, where both numbers are obtaine by computer search. The variables a DZ,ψ 2 an l DZ,ψ 2,i 2 are efine in a similar way while the error event starts in S D ψ an ens in S Z. Furthermore, t 1 specifies the number of cycles the trellis stays in φ an its upper limit makes sure that the total length of the unetectable ouble error oes not excee n ν, the number of trellis stages in the coewor. As in 5 P > ˆ,2 can often be neglecte because terms with 1 2 > ˆ have negligible probability. Although the number an lengths of error events connecting S Z an states in S D are obtaine by computer searches, the require search epth is only ˆ free. Moreover, a ZD,φ 1 l ZD,φ 1,i 1 an for all φ can be foun while searching for a ZZ an l,i ZZ because these two types of error events both start from S Z. Regaring a DZ,ψ 2 an l DZ,ψ 2,i 2, the associate error events start from 2 m 1 ifferent states an can be foun through 2 m 1 separate searches. Nevertheless, since these error events en at S Z, only one backwar search is necessary to capture all of them. In the backwar search, bits in shift registers move backwar. One can simply treat x as x 1 in polynomial representations an apply the same search algorithm. 3 Unetecte Error Probability: As shown in Fig. 4, an unetectable s-tuple error is compose of several parts: one error event from S Z to a state in S D, s 2 error events from a state in S D to a state in S D with visits to S N in between, the final error event from a state in S D to S Z, an also s 1 paths insie S D connecting consecutive error events. Note that in the example of Fig. 4, where s = 3, where the path connecting the first an the secon error events has a length of zero. Let φ u an ψ u be the inices of the start an en states of the u th transitions in S D, respectively. In Fig. 4, we have φ 1 = ψ 1 = 2 for the first transition; the secon transition starts from S D φ 2 an ens at S D ψ 2, where φ 2 = 3 an ψ 2 = 1. Also, let the number of error events starte at S D ψ u 1 an ene at S D φ u with istance u be a DD,ψu 1,φu u an the length of the i th u error event of them be l DD,ψu 1,φu u,i u, where both numbers are obtaine by computer search. Although we nee to perform 2 m 1 separate searches to obtain all a DD,ψu 1,φu u an l DD,ψu 1,φu u,i u, a search epth of ˆ s 1 free is sufficient. To simplify the notation, efine the following sets: Φ s = φ 1,, φ s φ u [1, 2 m 1] u [1, s]} Ψ s = ψ 1,, ψ s ψ u φu u [1, s]} [ [ I s = i 1,, i s i 1 1, a ZD,φ1 1 ], i s 1, a DZ,ψs 1 s ], [ ] } i u 1, a DD,ψu 1,φu u u [2, s 1] T s = t 1,, t s t u 0 u [1, s],

7 7 s φu t u n ν l min s1 where ls min is the shortest possible length of the unetectable s-tuple error specifie in a similar way as 14 an given by l min s = l ZD,φ1 s 1 1,i 1 l DD,ψu 1,φu u,i u u=2 s 1 s,i s l DZ,ψs 1 } δ φu,ψ u. 15 Similar to 7, the probability of having an unetectable error is now boune by P UD a ZZ 1 1 D ˆ,1 i=1 max 0, n ν l ZZ 1,i 1 } P 1 P s=2 1,, s D ˆ,s ψ 1,,ψ s 1 Ψ s 1 n ν l min s s u i 1,,i s I s φ 1,,φ s 1 Φ s 1 t 1,,t s 1 T s 1 s 1 φu t u 1 P > ˆ,s. 16 The last term is the probability sum of all large-istance errors, which are assume to be unetectable, an can be calculate using 11c. By letting P > ˆ,s = 0, we obtain an approximation. By letting all l ZZ, l ZD, l DD, an l DZ equal to ν 1, we obtain a looser boun which oes not require any length information. These two techniques are applicable to every P UD,s, incluing 12 an 13. C. Comparison The main benefit of the construction metho is that it is often able to search eeper than the exclusion metho because the output pattern is not require. However, the require memory size scales with the number of states 2 mν rather than 2 ν so this approach can encounter ifficulty in analyzing high-orer CRC coes. In contrast, the error events searche in the exclusion metho belong to the original convolutional coe, whose number of states is just 2 ν an is inepenent of the egree of the CRC coe. In fact, both methos create the ominant parts till or ˆ of the istance spectrum of the equivalent catastrophic convolutional coe with finite length. As explaine in Section IV, we foun it useful to raw on both approaches as we searche for optimal CRC polynomials for a specific convolutional coe. The exclusion metho is suitable for short-istance unetectable single errors, an the construction metho is preferre when we search for longer unetectable single errors an ouble errors. Fig. 5 compares the simulate unetecte error probability to the bouns prouce by the exclusion an construction methos. We consier the CRC coe px = x 3 x 1 concatenate with the memory size ν = 4 convolutional coe with generator polynomial 23, 35 8 in octal an free = 7. The information length is k = 1021 bits an thus the CRC coewor length is n = 1024 bits. The FER of this original convolutional coe is plotte as a reference. The equivalent, Probability Original Coe FER Equivalent Coe FER Unetecte Error Prob. P UD Exclusion P >14,s P UD Construction P >20,s E s /N 0 B Fig. 5. A comparison of the simulate unetecte error probability with the simulate frame error rate of the equivalent coe an the analyses from the exclusion an construction methos. The system is equippe with the CRC coe px = x 3 x 1 an the convolutional coe 23, The CRC coewor length is n = 1024 bits. catastrophic convolutional coe is 255, an its FER is also simulate. We can see from Fig. 5 that the unetecte error probability is upper boune by the FER of the equivalent coe, which equals P UD, the probability of having an unetectable error in the CRC coewor, an this boun gets tighter as SNR increases. The equivalent coe FER is above the probability of unetecte error because it is possible that a frame has both an unetectable error event an a etectable error event, which causes a frame error in the equivalent coe but oes not cause an unetecte error in the concatenate CRC an convolutional coes. The upper bouns of P UD are compute using the exclusion metho 7 an construction metho 16. In our calculation, the search epth limit of the exclusion metho is = 14, an ˆ = 20 is the epth limit for the construction metho. Since free = 7, only unetectable single an ouble errors are consiere. The probability sum of all large-istance terms given by 11c are plotte to verify that they are negligible, except for the exclusion metho at SNR below 0.75 B. Although the construction metho use a eeper search, it is still quite close to the exclusion metho even in the low SNR region. It can be seen that these analysis methos eliver accurate bouns at high SNR for both the unetecte error probability an the FER of the catastrophic coe. IV. SEARCH PROCEDURE FOR OPTIMAL CRC CODES In this section, we will present an efficient way to fin the optimal egree-m CRC coe for a targete convolutional coe an information length k. Note that the performance of a CRC coe epens on the information length [5]. A CRC coe may be powerful for short sequences but have numerous unetectable long errors that are prouce by a specific convolutional coe. Since the coefficients of x m an x 0 terms are both one, there are 2 m 1 caniates of egree-m CRC generator polynomials px. In principle, either the exclusion metho or

8 8 the construction metho can prouce the unetecte error probability for each caniate allowing selection of the best px. However, this process is time-consuming if m is large. Both exclusion an construction methos nee to compute the istance spectrum of unetectable errors up to some istance. We can reuce the computation time by skipping the istance spectrum searches of suboptimal CRC coes. When the FER is low, the unetectable error rate of a CRC coe is ominate by the unetectable errors with the smallest istance. Let the smallest istance of the unetectable errors be min. We can evaluate a CRC-convolutional concatenate coe by its istance spectrum at aroun min. To be more precise, a polynomial shoul be remove from the caniate list if it has a smaller min than the others or if it has more unetectable errors associate to the same min. In a convolutional coe, since the number of error events a grows exponentially as the associate istance increases, the cost to fin all unetectable errors grows exponentially as well. Hence, the CRC coe search starts with = free an upates the caniate list by keeping only the CRC generator polynomials with the fewest unetectable errors. Next, repeat the proceure with the next higher until only one polynomial remains in the list. When < 2 free, only single errors are possible. The exclusion metho can count the number of unetectable single errors of each caniate when. We can perform a computer search for error events of the original coe an check the ivisibility of each of them. Note that if an error event is foun to be unetectable, all of its possible offsets in the coewor shoul be counte. Since all caniates check the same set of error events, only one computer search with multiple ivisibility checks one for each CRC coe is sufficient. In contrast, using the construction metho requires construction of equivalent encoers for each caniate separately. Hence, for the initial values of near free, checking the ivisibility via the exclusion metho is preferre. Of course, once >, searching for unetectable single errors of the equivalent coes as in the construction metho is the only approach. When 2 free, unetectable ouble errors nee to be consiere in aition to single errors. The ivisibility test shoul be applie to all combinations of error event patterns e 1 x, e 2 x, an their gaps g 1. Even if the concept of cyclotomic cosets iscusse in Appenix A is utilize, we still nee to construct all cyclotomic cosets through about 2 m ivisions an also check if each of the remainer of e 2 x ivie by px is in the same cyclotomic coset as the remainer of e 1 x ivie by px. Alternatively, unetectable ouble errors can be irectly create using the construction metho. In the construction metho, the error events connecting S Z an states in S D with istances between free an free are foun through computer searches. In fact, these events can be generate using the etectable error events of the original coe previously foun by exclusion if free. For example, since the etectable error pattern e 1 x is known, the corresponing error event in the equivalent encoer trellis starts from S Z an traverses the trellis with the input sequence given by the quotient of x m e 1 x ivie by px. The state S D φ, where it ens, is thus etermine by the last m ν input bits, an its length l ZD,φ 1,i 1 has alreay been provie by the egree of e 1 x. For error events starting from states in S D an ening in S Z with pattern e 2 x previously obtaine by exclusion, traverse the trellis in reverse from S Z with the input sequence given by the quotient of x mνl e 2 x 1 ivie by x m px 1, where l = l DZ,ψ 2,i 2 1 is the egree of e 2 x. Note that x l e 2 x 1 an x m px 1 are the reverse bit-orer polynomial representations of e 2 x an px, respectively. The state S D ψ where the error event begins is etermine by the last m ν input bits in reverse orer. Accoring to the iscussion at the en of Appenix A, min is not likely to be much greater than 2 free when information length k is long enough. In other wors, the CRC coe search algorithm is usually finishe before reaching 3 free an oes not nee to count the number of unetectable triple errors. V. CRC DESIGN EXAMPLE FOR ν = 6, k = 1024 As an example we present the best CRC coes of egree m 16 specifically for the popular memory size ν = 6 convolutional coe with generator polynomial 133, with information length k = 1024 bits. Note that the propose esign metho is applicable to all convolutional coes an information lengths, an not limite to the choices use for this example. The corresponing unetecte error probability is also calculate an compare with existing CRC coes. Table II shows the stanar CRC coes liste in [5] an the best CRC coes foun by the search proceure in Section IV. For egrees with no stanar coes, those recommene by Koopman an Chakravarty in [5] are liste an calle K&C. The notation of generator polynomials is in hexaecimal as use in [5]. For example, CRC-8 has generator polynomial x 8 x 7 x 6 x 4 x 2 1 expresse as 0xEA, where the most an least significant bits represent the coefficients of x 8 an x 1 terms, respectively. The coefficient of x 0 term is always one an thus omitte. Table II also gives the istance spectrum of unetectable single errors a ZZ of each CRC coe up to = 22. The istance spectrum of the original convolutional coe a is given as a reference. Note that, since this convolutional coe has free = 10, a smaller a ZZ 20 or a ZZ 22 oes not mean fewer unetectable errors at istance = 20 or 22. Unetectable ouble errors shoul also be counte for 20 to juge a caniate. During the search for the best CRC coes with egrees m 11, only single errors nee to be consiere because one caniate will outperform all the others before looking at = 20. Although the best egree-11 CRC coe has min = 20, all the other caniates have min < 20 an are eliminate before the en of the = 18 roun. Since the lengths of single errors l,i for < 20, ranging from 7 to 43, are much shorter than n ν, a caniate that has fewer types of ominant unetectable error events will have fewer ominant unetectable errors in total. In other wors, when unetectable single errors ominate an information length k is long enough, the best CRC coe shoul possess the smallest a ZZ. When min 2 free, the ominant unetectable errors inclue ouble errors. In this case, a smaller a ZZ oes not mean a better coe because it only consiers single errors.

9 9 TABLE II STANDARD CRC CODES OR CRC CODES RECOMMENDED BY KOOPMAN AND CHAKRAVARTY K&C [5], AND THE BEST CRC CODES FOR CONVOLUTIONAL CODE 133, WITH k = 1024 BITS. Gen. Unetectable Single Distance Spectrum a ZZ Name Poly K&C-3 0x Best-3 0x CRC-4 0xF Best-4 0xD CRC-5 0x Best-5 0x CRC-6 0x Best-6 0x CRC-7 0x Best-7 0x CRC-8 0xEA Best-8 0x K&C-9 0x Best-9 0x CRC-10 0x Best-10 0x CRC-11 0x5C Best-11 0x CRC-12 0xC Best-12 0xA K&C-13 0x102A Best-13 0x1E0F K&C-14 0x21E Best-14 0x314E K&C-15 0x Best-15 0x604C CRC-16 0xA Best-16 0x8E Original Distance Spectrum a For example, the egree-16 polynomials 0xF8F1 an 0x8E61 both have min = 22. The former has a ZZ 22 = 0 while the latter has a ZZ 22 = 1. However, at = 22, the former has so many 2860 unetectable ouble errors that the number is greater than the total count of unetectable single an ouble errors of the latter, when the information length is k = 1024 bits. However, when k = 512 bits, the former has fewer unetectable errors an becomes optimal. Therefore, ifferent information lengths may lea to ifferent optimal CRC esigns. The authors of [5] have inclue the information length k as a key esign parameter an propose a methoology to etermine goo CRC coes for a range of k. The same rule can be applie here. First, fin the CRC polynomials possessing the largest min for the longest k. Then, consier shorter k an keep the CRC polynomials having the largest min, which might increase as k ecreases. Table III shows the best CRC coes for k = 256, 512, or 1024 bits an ientifies the goo coes for this range of information lengths. The bol numbers inicate the k for which the coe is esigne. For example, the coe 0xA10 is the best egree-12 coe at lengths k = 256 an 1024 bits. Note that the coes with egrees m 11 are not shown since unetectable single errors ominate an thus the best CRC coes for these k are ientical. In fact, the best CRC coe for the largest k is usually goo for shorter k. In our case, TABLE III THE BEST CRC CODES FOR CONVOLUTIONAL CODE 133, WITH k = 256, 512, OR 1024 BITS. BOLD NUMBERS INDICATE THE k FOR WHICH THE CODE IS DESIGNED. THE GOOD CODES FOR THIS RANGE OF k ARE SPECIFIED. Unetecte Error Probability Gen. min, count of unetectable errors at min Degree Poly. k 256 bits 512 bits 1024 bits xA10 goo 20, , , x8DC goo 20, , , x18F6 goo 20, , , x1E0F goo 20, , , x2E20 22, , , x314E goo 22, , , 198 0x6D80 22, , , x76AD 22, , , x604C goo 22, , , xA219 24, , , xF8F1 goo 24, , , x8E61 22, , , 2435 Conv. FER CRC-4 CRC-6 CRC-8 CRC-10 CRC-12 K&C-14 CRC-16 Best-4 Best-6 Best-8 Best-10 Best-12 Best-14 Best P >28,s 12 E s /N 0 B Fig. 6. The unetecte error probability of the existing an best CRC coes for convolutional coe 133, with information length k = 1024 bits compute using the construction metho. the best k = 1024 CRC coes have no more than 0.1B loss compare to the best k = 256 an the best k = 512 CRC coes at information lengths k = 256 an 512 bits, respectively, except for the egree-16 coes. In Fig. 6, the bouns of unetecte error probability of the existing an best CRC coes for information length k = 1024 bits are shown. For clarity, only even egrees of the CRC coes are isplaye. The upper boun of the original convolutional coe FER without any CRC coe, calculate using transfer function techniques [14], is plotte as a reference. In our calculation, the search epth limit of the exclusion metho is = 22 an not enough for high egree CRC coes. Therefore, the construction metho 16 was use with ˆ = 28. The probability sum of all large-istance terms calculate using 11c is also plotte to illustrate that the large-istance terms really are negligible even assuming they are all unetectable, except for the best egree-16 CRC coe at SNR below 7 B an some other coes at SNR below 6 B. Note that since the operation e P 1 in 11c causes non-negligible rouning errors in the high SNR region, it was approximate

10 10 Unetecte Error Probability Existing CRC Coes Best CRC Coes CRC Length bit Require CRC Length bit Existing CRC Coes Best CRC Coes E s /N 0 B Fig. 7. The unetecte error probability of the existing an best CRC coes for convolutional coe 133, with information length k = 1024 bits at SNR = 8 B. Fig. 8. The require CRC lengths for the existing an best CRC coes for convolutional coe 133, with information length k = 1024 bits to achieve unetecte error probability below by the first ten terms, which results a similar expression as 11b but only carrie out for 1 s 10. Since ˆ < 3 free, it is not necessary to evaluate triple or higher orer errors uner this truncation. It is clearly seen from the figure that the best egree-m CRC coes foun by our proceure outperform the existing egree-m coes for all m. Furthermore, their performance is either better than or similar to the existing egree-m 2 coe except for m = 6. In other wors, the propose esign can typically save 2 check bits while keeping the same error etection capability. We can compare the error etection capability of all coes at a fixe SNR. If we raw a vertical line at SNR = 8 B on Fig. 6, the intersections can be plotte along with the associate CRC lengths m in Fig. 7. The largest reuction of unetecte error probability is about five orers of magnitue at m = 5 where the existing CRC coe has an unetecte error probability of an the newly esigne CRC coe has an unetecte error probability of , base on the analysis. Note that these numbers are calculate at 8 B SNR, an the reuction gets more significant as SNR increases. Since the existing CRC coes are not tailore to the convolutional coe, a higher egree coe oes not necessarily have a better performance. We observe that there are three almost horizontal region for the existing CRCs with egrees m ranging from 3 to 5, from 8 to 9, an from 13 to 15. The reason is that the coes in each region have similar number of ominant unetectable errors as can be seen in Table II. In contrast, the CRC coes esigne using our proceure show steay improvement as the egree increases. The existing an best CRC coes can also be compare in terms of the require CRC length to achieve a certain unetecte error probability. Assume our target is to reach unetecte error probability below This can be shown by rawing a horizontal line at the target probability level on Fig. 6 an plotting the CRC lengths associate to the crosse points as a function of SNR as in Fig. 8. For most of the SNR levels, the best CRC coes requires two fewer check bits than the existing CRC coes to achieve the same error etection capability. At SNR aroun 10.5 B, the best CRC coe requires three fewer check bits than the existing CRC coe. Since the existing coe require six check bits, this is a 50% reuction. VI. CONCLUSION A goo CRC coe in a convolutionally coe system shoul minimize the unetecte error probability. To calculate this probability, two methos base on istance spectrum are propose. The exclusion metho starts with all possible single an multiple error patterns of the convolutional coe an exclues them one by one by testing if they are etectable. In the construction metho, unetectable errors are mappe to error events of an equivalent convolutional coe, which is the combination of the CRC coe an the original convolutional coe. The computer search for error events in the construction metho oes not nee to recor the error patterns an thus can go eeper than the search in the exclusion metho. However, the construction metho coul encounter ifficulties while ealing with high-egree CRC coes. Moreover, the construction metho is generally applicable to the performance analysis of catastrophic convolutional coes. We also propose a search proceure to ientify the best CRC coes for a specifie convolutional encoer an information length. A caniate CRC coe is exclue if it has more low-istance unetectable errors. Therefore, the best CRC polynomial is guarantee to have the fewest ominant unetectable errors an minimizes the probability of unetecte error when SNR is high enough. When unetectable ouble errors ominate, the choice of the best CRC polynomial is more epenent on the information length. In an example application of the esign proceure for the popular 64-state convolutional coe with information length k = 1024, new CRC coes provie significant reuction in unetecte error probability compare to the existing CRC coes with the same egrees. With the propose esign, we are able to save two check bits in most cases while having the same error etection capability. It is an open problem to generalize this work to other errorcorrecting coes, such as turbo or low-ensity parity-check LDPC coes. The construction metho can only be applie when the encoer has the convolutional structure. We note that convolutional LDPC coes have such a structure. For turbo coes, the construction metho can be use to analyze one of the recursive systematic convolutional RSC coes but analyzing the other RSC coe is not straightforwar ue to the existence of the interleaver. Nevertheless, the exclusion metho can be generalize to any other error-correcting coes