the following description is intended to convey a thorough understanding of the invention by providing a number of specific embodiments and details involving the structure and operation of a novel method and system of the present invention . it should be understood , however , that the present invention is not limited to these specific embodiments and details , which are provided for exemplary purposes only . it should be further understood that one possessing ordinary skill in the art , in light of known and methods and system , would appreciate the use of the invention for its intended purposes and benefits in any number of alternative embodiments , depending upon specific design and other needs . fig2 is bit - loading method and system according to at least one embodiment of the present invention . conventional bit - loading algorithms are based on the shannon channel capacity formula to calculate the bits that a discrete multitone modulation ( dmt ) system can support , and may be calculated using equation ( 2 ) as follows : b dmt = i . times . b i = i . times . log . function . ( 1 + snr i . times . c / m . times . . times . . gamma .) ( 2 ) where b . sub . dmt is the total bits that a dmt system may support , b . sub . i is the bits supported by the ith sub - channel , snr . sub . i is the snr in the ith sub - channel , c is the coding gain , the summation is over all the sub - channels used and m is the margin , an amount of additional signal - to - noise ratio in excess of the minimum required to achieve a given performance level for a particular type of modulation scheme with a particular snr gap . gamma . thus , for all the sub - channels used , the coding gain and the gap are common . reference is made to fig2 . in one embodiment of the present invention , a method and system for a bit - loading algorithm is disclosed , where the snr gap and the coding gain varies and where both depend on the constellation size ( or the bits supported ), and the mapping method employed . since dmt is a combination of different constellation sizes , the method and system for loading bits in sub - channels is calculated using equation ( 3 ) as follows : b dmt = i . times . b i = i . times . log . function . ( 1 + snr i . times . c i / m . times . . times . . gamma . i ) ( 3 ) that is , the systems and methods according to the various embodiments of the present invention take into consideration a variable coding gain c . sub . i and a variable snr gap . gamma .. sub . i for different sub - channels with different constellation sizes , as depicted in the steps 210 , 220 , 230 , 240 , 250 , 260 shown in fig2 . another advantage of the systems and methods of the present invention is that bits are allocated to sub - channels or “ bins ” based on an equal error rate principle for each sub - channel in the dmt system . as a result , efficiency gains are realized because it is a “ per sub - channel ” analysis of the error rate , and it is iterative in the sense that a first guess leads to an error per sub - channel in the frequency domain , which is then refined only once leading to a final dmt symbol estimation . assuming equal and optimal system performance margin over all used sub - channels at an specified error rate , the probability of error ( symbol error ) in each sub - channel i , may be calculated by equation ( 4 ) as follows : p . sub . s = k ( i ) q ( d ( i )/ 2 . sigma .) ( 4 ) where k ( i ) is the nearest symbol neighbor for the ith sub - channel , d ( i ) is the corresponding minimum distance , σ is the standard deviation , and . sigma .. sup . 2 is the 1 - dimensional additive white gaussian noise ( awgn ) variance . then the minimum is achieved when all sub - channels have the same error rate . in this way , any sub - channel will have an equal chance to have an error and the error probability is evenly distributed to each sub - channel . for a dmt with awgn , the channel capacity for any sub - channel is calculated as follows : b = log . sub . 2 ( 1 + snr . sub . c ) ( 5 ) where snr . sub . c is the received signal - to - noise ratio per 2 - dimensions . in other words , the snr required to support a given number of bits per sub - channel in this ideal case is : snr . sub . c = 2 . sup . b − 1 ( 6 ) however , the actual snr required to support a required rate is calculated from equation ( 4 ) to approximate for large snr , specifically for the un - coded case , where there is an absence of an error - correction coding scheme involving the implementation of redundant bits and expanded bandwidth . as such , the actual snr may be calculated by equation ( 7 ) as follows : snr r = a [ q − 1 ( p b / k )] 2 ( 7 ) where a is a number related to the size and shape of the constellation , p b is the required bit error rate , q is the error function , and k is the nearest symbol neighbor . the snr gap , the difference between the actual snr and the received signal snr , may be calculated by equation ( 8 ) as follows : the snr gap γ is a factor that represents the effects of the chosen coding scheme , the target bit error rate ( ber ), and the desired minimum system performance margin . in particular , the snr gap effectively estimates the difference between channel capacity and the actual capacity usage of the transmission scheme . and from ( 8 ) above , γ is not constant in dmt . rather , it is a parameter that depends on the shape and the number of bits allocated to each sub - channel . quadrature amplitude modulation ( qam ) is the encoding of information into a carrier wave by varying the amplitude of both the carrier wave and a quadrature carrier which is 90 . degree . out of phase with the main carrier in accordance with two input signals . qam takes advantage of the fact that the greater the number of symbols , the greater the efficiency of the system . occupied bandwidth is determined , mostly , by the symbol rate . so the more bits ( the fundamental information units ) per symbol , the higher the efficiency . fig3 is a qam signal with a 4 - point signal constellation according to at least one embodiment of the invention . fig4 is a qam signal constellation of 8 points according to at least one embodiment of the invention . the number of symbols required for a given system is 2 . sup . n , where n is the number of bits per symbol . for a 16 qam , n = 4 and there are 16 symbols — each symbol representing four bits : 0000 , 0001 , 0010 , etc . symbol error probability is difficult to ascertain , therefore , a union bound technique is almost always used . the union bound is an upper bound on the symbol error probability at large snrs ( i . e . for good communication channels ). for a qam system in awgn , the symbol error rate may be calculated using a union - bounded ( upper limited by the sum of all the errors made to all the wrong decision at all different differences ) as in equation ( 9 ) as follows : p s . ltoreq . i = 1 n . times . . alpha . i . times . q . function . ( d i / 2 . times . . sigma .) ( 9 ) where . alpha .. sub . i is the average symbol error multiplicity for two constellation points y . sub . n and y . sub . m with squared distance d . sub . i . sup . 2 , and the q function is the pair - wise error probability . from ( 9 ), it is obvious that for a large snr with un - coded modulation , only the first few terms are important , which means a large enough minimum distance between the constellation points is required . for un - coded systems , only the first term ( at most the first two ) is of importance in the case of a large snr . this is also true even for coded systems , for instance , in trellis - coded modulation systems . however , for middle to low snr , the optimization of the weight distribution needs to be considered . that means more terms in equation ( 9 ) are needed for the bit error rate calculation . for applications such as dmt , it is the bit error rate p b that is of greater importance . usually p b & lt ; p s for a qam signal with bits / sub - channel , where b & gt ; 1 . in this case , when a decoded symbol is in error , not all the corresponding bits are in error . the union bound for bit error rate p b is represented by equation ( 10 ) as follows : p b ≤ ∑ i = 1 n ⁢ β i ⁢ q ⁡ ( d i / 2 ⁢ σ ) / b ( 10 ) where β i is the average bit error multiplicity at distance d i and it is defined as β i = ∑ n ⁢ ∑ m ⁢ pr ⁡ ( y m ) ⁢ β ⁡ ( m , n ) ( 11 ) where pr ( y m ) is the probability of constellation point y m and β ⁡ ( m , n ) = { h ⁡ ( x ⁡ ( y n ) , x ⁡ ( y m ) ) , d ⁡ ( y m , y n ) = d i 0 , ⁢ otherwise ( 12 ) h is a function returning the hamming distance between two b - tuples , x ( y m ) and x ( y n ) mapping to y m and y n , according to the dmt standard . in an embodiment of the present invention , and in reference to fig5 , a minimum distance d min of 2 for a qam constellation is utilized , in order to show that only the first two terms of the distance spectrum is necessary for the ber calculation . the distance spectrum for un - coded dmt utilizing d min of two for a qam constellation yields the squared distances and the corresponding symbol and bit multiplicities at those distances for the next reference ( fig6 ). the union bound by its nature is a very loose bound , and to get a tight bound , some form of truncation has to be implemented . for the snr range of interest in dmt which is greater than 14 db for 2 bits per sub - channel in g . dmt or greater than 11 db for 1 bit per sub - channel in g . dmt . bis for the un - coded case , only the first term in equation ( 10 ) is dominant due to the q function and the large distances for other terms ( the q function will drop exponentially as the distance increases ). the contribution from the second term is far less than that from the first term for snr larger than 10 db . therefore , the bit error rate is approximated by equation ( 13 ) as follows : p b = β 1 q ( d min / 2σ )/ b ( 13 ) where d min is the minimum distance among the constellation points for a given constellation for the un - coded case . furthermore , the systems and methods of the present invention examine the bit error rate for the whole dmt system . if the ith sub - channel has a ber p b , i and can support b i bits , the contribution of this sub - channel to the total ber is where the sum is for all the sub - channels used . thus , the ber for the dmt system is represented by equation ( 14 ) as follows : for an optimum system design , all sub - channels should perform with the same bit error rate , otherwise , the sub - channel with the highest bit error rate would dominate . for a dmt system , at least a bit error rate of p b = 10 − 7 is required according to the dmt standards . in addition , the probability of a decoding error is reduced by incorporating gray coding in the dmt signaling , a coding scheme where adjacent symbols differ by 1 digit . this type of coding is useful for reducing the bit error rate in the dmt system . also the criterion of equal bit error rate is not the same as the criterion of equal symbol error rate for the dmt system design because of the difference between equations ( 9 ) and ( 10 ) ( one symbol will typically introduce more than 1 bit error ). according to an embodiment of the present invention , and in accordance with fig6 , the snr gap being related to different criteria , is calculated , and a table is created according to equation ( 8 ). generally , the snr gap is calculated such that for p s = 10 − 7 , the snr gap is 9 . 8 db or for a p b = 10 − 7 , the snr gap is 9 . 5477 db , but actually , the 9 . 8 db ( according to the table , for a 2 bits per bin , the gap is actually 9 . 758 db ) or the 9 . 5477 db is only for the case of 2 bits per sub - channel , and not across all the dmt sub - channels having different bits . the gap is not a constant for all cases , especially for 1 bit per sub - channel and 3 bits per sub - channel . instead , the gap depends on the bits supported by a constellation . this is one of the new results in this invention . therefore , we can see that the channel capacity formula ( 5 ) and a single snr gap value ( say 9 . 5477 db ) are not good for the system design . instead , different values of the snr gap for the different bits supported by a sub - channel must be used . for the coded case , the trellis encoder / decoder in dmt comes from wei &# 39 ; s original 4d , 16 state , rate ⅔ trellis coded modulation scheme . the coded case shares many parameters with the un - coded case , but at the same time there are enough differences that may introduce difficulty in the performance evaluation for the dmt receiver . for example , in wei &# 39 ; s original code , if there is a requirement to transmit q information bits per 2d in 1 symbol interval , 2q + 1 bits in one 4d constellation should be transmitted . in other words , in each symbol interval , transmitting ( 2q + 1 )/ 2 bits per 2d ( which may be fractional or integer ) is possible . but in any case , the same bits in each symbol interval must be sent in a per 2d . therefore , the coding gain may be easily calculated because the comparison is always done between a 2d constellation with size 2 ( 2q + 1 )/ 2 and that with size 2 q ( q may not be an integer which facilitates much flexibility for comparison of the coded and un - coded cases because the constellation points are not confined to the power of 2 . on the other hand , in dmt , if bit 1 + bit 2 − 1 information bits are allocated to sub - channels 1 and 2 for transmission , two of the bit 1 + bit 2 − 1 information bits are sent to the input of the convolutional encoder , and three bits are outputted from the encoder , which are then used to select 1 of an 8 4d subsets . the remaining information bits are used to select point in the 4d subset . because bit 1 + bit 2 are sent for the 2 sub - channels , it can be seen that the coding gain calculation will become difficult because bit 1 and bit 2 are always integers and may not be always equal , even though tone - ordering is utilized . other drawbacks exist in calculating of the bit error rate in dmt . because two sub - channels constitute one 4d constellation , any error in one sub - channel will influence the errors in other sub - channels , dmt utilizes only one convolutional encoder across all the sub - channels in a one dmt symbol , essentially resulting in a time - varying encoder , because the constellation sizes vary from sub - channel to sub - channel . for instance , from any given time n to n + 1 , there may be 2 parallel transitions . but from time n + 1 to n + 2 , there might be as much as 128 or more parallel transitions . recalling that a first error event includes more than one branch metric difference in trellis coding , the error rate in one sub - channel is actually coupled to other sub - channels . from this point of view , equation ( 15 ) above , yields the direction and the best choice is of course p b , dmt = p b . in addition , wei only considered an infinite lattice and ignored the boundary effect . at the same time , he only paid attention to the kissing number ( in geometry , the kissing number is the maximum number of spheres of radius 1 that can simultaneously touch a unit sphere in n - dimensional euclidean space . the kissing number in n dimensions , with n = 4 , is 24 ) for the 4d symbol but not the bit error multiplicity , thus there are 24 such kissing numbers per 4d and each of them comes from the 4d symbol detection error at the intra - sub - lattice minimum squared euclidean distance — msed . however , in practice , not an infinite lattice , but a finite lattice is used to construct a 4d code . that means that the boundary effect must be considered , which turns out to be vital for dmt system design . on the other hand , it is the system bit error rate ( related directly to the bit error multiplicity ) that is of most interest , because it is this error rate that determines the final ber . the boundary effect and the bit error multiplicity plays a very important role in determination of the performance for the case without trellis - coded modulation ( tcm ). for the case with tcm , its role will be more interesting because in the coded case , a dense lattice rd 4 that is a rotated version of lattice d 4 is used , where the kissing number is 24 for rd 4 ( per 4d ). therefore , a stronger impact of multiplicity ( or error coefficient ) on the performance for the coded case should be observed . it is very hard to analyze the error performance in dmt with tcm due to the time - varying property and the coupling between different sub - channels . however , a general formula can be provided for the ber in a particular sub - channel from equation ( 10 ) as shown in equation ( 15 ) as follows : where p b , para is bit error rate introduced by parallel branch in the trellis diagram which is dominated by the intra sub - lattice msed or d min ( the minimum squared distance for rd 4 in tcm in dmt is related to the un - coded minimum distance by d min 2 = 4d 1 2 ). on the other hand , the second term p b , unpara in the right side is the bit error rate from the un - parallel transition , which is mainly contributed to the free distance d free of the trellis . in wei &# 39 ; s rate ⅔ , 16 state trellis code , this free distance is larger than the intra sub - lattice minimum msed . the first term in the right side of equation ( 16 ) below can be approximated by equation ( 14 ). the second term is bounded by : p b , unpara ≤ 1 2 ⁢ b ⁢ q ⁡ ( d free 2 / 2 ⁢ n 0 ) ⁢ exp ⁡ ( d free 2 / 4 ⁢ n 0 ) ⁢ ∂ ∂ z ⁢ t ⁡ ( w , z )  z = 1 , w = exp ⁡ ( - 1 / 4 ⁢ n 0 ) ( 16 ) due to the face that the convolutional encoder is being across all the sub - channels that are not with the same constellations , we must at least guarantee all the parallel paths have to have the same bit error rate along the trellis . and because the free distance d free of the trellis is larger than the minimum distance d min for lattice rd 4 , and in dmt the snr is normally high to keep the ber p b = 10 − 7 , the contribution from p b , unpara is not considered . therefore , as in the un - coded case , it is the bit error multiplicity that determines the ber . for this reason , both the symbol and bit error multiplicities are shown in the table fig7 for the final coding calculation . it should be noted that in real dmt system , even though tone ordering is offered , given the bits for first constituent sub - channel , the second sub - channel does not have the same constellation size in the same 4d symbol . to simplify the problem , the method and system of the present invention utilizes the same bits for both sub - channels in tabulating the table elements of fig7 . the second column of the table is the average nearest neighbors for different bits / sub - channel . the number is normalized to per 2d constellation . it is clear that when the constellation size becomes large , the symbol error multiplicity ( the kissing number ) at msed = 16 ( d 1 = 2 ) is clase to 12 . on the other hand , in 4 qam case , there is only 1 kissing number per 4d ( instead of 24 for an infinite lattice ). actually , because one 4d symbol is composed of two 2d constellations ( see fig3 and 4 for examples of 2d constellations ) dividing the 4d symbol into 8 4d subsets will reveal the nearest neighbors . a rule of thumb is that every doubling of the kissing number will cost a 0 . 2 db snr reduction so , a difference near 1 db in the snr required between 2 2 and 2 15 constellation points per sub - channel ( per 2d ) will be seen in terms of symbol error rate . therefore , the boundary effect must be considered in the coded case for the coding gain calculation . several numerical examples may give us more insight about the boundary effect . for 2 bits / sub - channel , 4d point ( 0 , 0 ) has only 1 nearest neighbor ( corresponds to 4 bits in difference with the neighbor ). for 4 bits / sub - channel , 4d point ( 0 , 0 ) has 20 nearest neighbors ( corresponds to 100 bits in difference ). but 4d point ( 3 , 3 ) has only 8 neighbors in msed ( corresponds to 24 bits in difference ). and for 5 bits / sub - channel , 4d point ( 0 , 0 ) has 24 nearest neighbors ( the boundary effect is gone and there are 106 bits in difference with the neighbors ). but 4d point ( 24 , 24 ) has only 18 nearest neighbors . in any constellation , a real inner point will have a kissing number 24 but the outer points will never reach this number . the main difference between a small constellation and a large one is that the former has more percentage for the outer points and therefore the boundary effect is more obvious . for a single - carrier system , generally only one constellation is dealt with . but for a system like dmt , many qam symbols join together and this big difference must be taken into consideration , especially for the case of tcm . the third column in the table of fig7 presents bit error multiplicity at the intra sub - lattice msed . as the bits per sub - channel increase , the bit error coefficient for this sub - channel becomes large , which is in agreement with intuition because a large constellation means more average neighbors at msed and small boundary effect . comparing the coded and un - coded cases , columns 5 and 6 give the same numbers as columns 2 and 3 for un - coded case . it can be seen that both the symbol and bit error multiplicities are much smaller for un - coded case due to the sparser lattice . particularly for dmt , and in reference to columns 4 and 7 of the table of fig7 , the method and system of the present invention tabulates the average error bits in one 2d symbol for both the coded and un - coded cases . it can be seen from the tabulated numbers that for the coded case , the number is between 3 to 7 but for the un - coded case , it is between 1 and 3 , this indicates that if 1 2d symbol error is made , 3 to 7 bit errors will be generated for the coded case . however , there are only 1 to 3 bit errors per 2d symbol error for the un - coded case . according to the relationship between the symbol error and bit error in the coded case , and the viterbi decoding of wei &# 39 ; s 4d trellis code , when a decoder error is made , it is a 4d symbol error . such an error may introduce at most 3 bit errors ( recall that only 3 bits are coming out from the convolutional encoder for one 4d symbol , which select one of 8 4d sub - lattices ). but according to the method and system of the present invention , there may be a 4d symbol error without any coded bit error . a simple example is that a received 4d point is crossing the detection boundary in the same 4d sub - lattice , there will be un - coded bit errors but there is no coded bit error . the second column in the table of fig8 is the snr to support a certain constellation in the coded case , which is of course smaller than that for the un - coded case . the snr for the un - coded case is the sum of column 2 and 3 . the difference between the two is the nominal coding gain , which is offered in third column . as such , it can be seen that the coding gain in dmt is not a constant , as given in wei &# 39 ; s code . c no min al = 10 log 10 ( d min 2 / d 1 2 )= 6 . 0206 ( db ) ( 17 ) rather , it depends on the bits supported by the sub - channel ( i . e . the constellation ). the method and system of the present invention , and in reference to the table of fig8 , discloses that for dmt systems , due to the large error coefficient for a large constellation ( say 11 bits / sub - channel ), the coding gain is smaller compared to that of a small constellation ( say , 2 bits / sub - channel ), as shown in the tabulated coding of 6 . 0206 for 2 bits / sub - channel as compared to 5 . 3874 for 11 bits / sub - channel in column three of the table of fig8 . this nominal coding gain is achieved without considering the influence of the constellation expansion due to the trellis coding . by reducing the nominal coding gain by 1 . 5 db , the effective coding gain is provided in column 4 . for the 2 bits / sub - channel case , the result is the same as wei &# 39 ; s . however , for other bits / sub - channel , less coding gain is attained . the value off from 4 . 5206 db ( 2 bits / sub - channel ) is shown in the fifth column . it can be seen that at 15 bits / sub - channel , this difference is close to 0 . 62327 db . therefore , in the actual system implementation , this should also be considered ( the actual number may change if the non - parallel paths are also put into consideration ). in literature , the rate supported by a given dmt system is calculated according to equation ( 3 ), with a fixed trellis coding gain c as follows : b dmt = ∑ i ⁢ b i = ∑ i ⁢ log ⁡ ( 1 + snr i ⁢ c / m ⁢ ⁢ γ ) ( 3 ) where the summation is over all the sub - channels used m is the margin and γ is the gap . that is , for all the sub - channels used , the coding gain and the gap are common . but from the discussion above , the gap and the coding gain are both depending on the bits that the sub - channels can support . therefore , to sustain a given bit error rate , we have to lose some snr for some sub - channels with lower coding gain or a higher gap according to equation ( 3 ). but from the discussion above , the method and system of the current invention has shown that due to the varying snr gap and the varying coding gain , the rate supported by a given dmt system may be calculated using equation ( 2 ) as follows : by utilizing the tabulated snr gap of the table of fig6 and the tabulated snr requirement for tcm of the table of fig8 above , bits can be allocated to dmt sub - channels with greater efficiency resulting in a much better performance than the current bit - allocation methods and systems . the present invention is not to be limited in scope by the specific embodiments described herein . indeed , various modifications of the present invention , in addition to those described herein , will be apparent to those of ordinary skill in the art from the foregoing description and accompanying drawings . thus , such modifications are intended to fall within the scope of the following appended claims . further , although the present inventions have been described herein in the context of a particular implementation in a particular environment for a particular purpose , those of ordinary skill in the art will recognize that its usefulness is not limited thereto and that the present invention can be beneficially implemented in any number of environments for any number of purposes . accordingly , the claims set forth below should be construed in view of the full breath and spirit of the present inventions as disclosed herein .