To compute real GDP by state on NAICS (1997-forward), you first need the time series of Quantity Indexes for Real GDP by State. Next, divide each quantity index by 100 to put these on a basis of 1.0. Then, multiply each quantity index by the base year GDP-by-state value (you can use current or real dollars since these values are the same in the base year). The result will be the real GDP-by-state series.
To compute real GDP by state on SIC (1977-1997), requires a slightly different calculation because the base year values on SIC are not available. You first need the time series of Quantity Indexes for Real GDP by State. Next, divide the quantity index for the year you are trying to compute, for example 1977, by the quantity index of one of the years for which real GDP-by-state data is available, for example 1990. Then, multiply the result times the real GDP-by-state value for the year available, in this example 1990.
These methods work because the movement in the two data series (Real GDP by state and the Quantity Indexes for Real GDP by State) is the same—the percent change in one is equal to the percent change in the other.
You should use these constructed Real GDP-by-state values only for the unpublished years, and not to replace the actual published values. It is a good idea to use this method to compute a published real GDP-by-state value just to verify that you have the algebra working correctly. You won't hit the published number exactly, because the published values are computed using un-rounded data, but you should come very close.